On the discretised ABC sum-product problem

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC 4.0 https://creativecommons.org/licenses/by-nc/4.0/ On the discretised ABC sum-product problem © 2023 the Authors Accepted version (Final draft) Orponen, Tuomas Orponen, T. (2024). On the discretised ABC sum-product problem. Transactions of the American Mathematical Society, Early online. https://doi.org/10.1090/tran/9094 2024 arXiv:2110.02779v3 [math.CO] 10 Nov 2023 ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM TUOMAS ORPONEN ABSTRACT. Let 0ăβďαă1and κą0. I prove that there exists ηą0such that the following holds for every pair of Borel sets A, B ĂRwith dimHA“αand dimHB“β: dimHtcPR: dimHpA`cBq ď α`ηu ď α´β 1´β`κ. This extends a result of Bourgain from 2010, which contained the case α“β. The paper also contains a δ-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a1B`...`anB. CONTENTS 1. Introduction 2 1.1. Related work 4 1.2. Comparison to classical projection theorems 7 1.3. Paper outline and proof sketch 7 1.4. Acknowledgements 9 2. Notation and preliminaries 9 2.1. Dyadic cubes and covering numbers 9 2.2. Entropy 9 3. Three initial reductions 11 3.1. Reduction to the case where Bhas small doubling 11 3.2. Reducing the Frostman constant of ν14 3.3. Removing reference to subsets 17 3.4. Bonus reduction 20 4. Proof of Theorem 3.28 21 4.1. Preliminaries 21 4.2. Shmerkin’s inverse theorem 21 4.3. Applying the inverse theorem 23 4.4. Pruning B1to improve separation I 25 4.5. Intervals with small but non-zero B1-branching 26 4.6. Branching of A1on typical intervals in N`29 4.7. Pruning B1to improve separation II 30 4.8. Elementary projection estimates 32 4.9. Projecting pieces of A1ˆB234 4.10. Final multiscale argument 36 Date: November 13, 2023. 2010 Mathematics Subject Classification. 11B30 (primary) 28A80 (secondary). Key words and phrases. Discretised sum-product problem, Projections, Hausdorff dimension. T.O. is supported by the Academy of Finland via the projects Quantitative rectifiability in Euclidean and non-Euclidean spaces and Incidences on Fractals, grant Nos. 309365, 314172, 321896. 1 2 TUOMAS ORPONEN 5. Hausdorff dimension estimates 38 5.1. Reducing Theorem 1.6 to Theorem 1.8: outline 38 5.2. A toy version 39 5.3. Reduction to a weaker toy theorem 39 5.4. Proof of the weaker toy theorem 41 5.5. Proof of the main theorem 47 5.6. Proof of Corollary 1.7 51 References 54 1. INTRODUCTION Let A, B, C ĂRbe large but finite sets. Is it true that there exists some cPCsuch that |A`cB| " |A|? Here |¨|refers to cardinality. Not necessarily: consider for example An“!1 n1{2,2 n1{2,...,1)and Bn“!1 n1{4,2 n1{4,...,1)“Cn.(1.1) It is not hard to check that for every ǫą0, there exists nPNsuch that |An`BnCn| ď nǫ|A|, so in particular |An`cBn| ď nǫ|A|for all cPCn. The problem can be fixed by adding one assumption: |B||C| " |A|. Then, a positive answer to the question follows easily from the Szemerédi-Trotter theorem [39] applied to the planar set AˆB. The requirement |B||C| " |A|is also necessary, as one can see by variants of (1.1). The ABC sum-product problem, stated above, also makes sense in contexts where the Szemerédi-Trotter bound is not available, for example if A, B, C ĂZp, and pPN is prime. Again, it turns out that the lower bound |B||C| " |A|yields the existence of cPCwith |A`cB| " |A|. One way to show this is to adapt elementary techniques of Garaev [11], Glibichuk and Konyagin [12], and Bourgain [4]. The details can be found in [29]. Another way is to apply directly an incidence bound in finite fields due to Stevens and de Zeeuw [38]. The theorem of Stevens and de Zeeuw gives a stronger lower bound for |A`cB|than the elementary approach (see [29, Proposition 1.3] for the details), but ultimately relies on the polynomial method. The purpose of this paper is to consider the δ-discretised ABC sum-product problem in R, and lower bounds for dimHpA`cBq, the Hausdorff dimension of A`cB. The δdiscretised problem is otherwise the same as the question we started with, but instead of counting the cardinality |A`cB|, we seek lower bounds for the δ-covering number |A`cB|δfor some small scale δą0. We will also assume that the sets A, B, C are δseparated, and have cardinalities |A| “ δ´α,|B| “ δ´β, and |C| “ δ´γ. In this variant of the problem, hypotheses on |B||C|need to be coupled with additional non-concentration conditions to hope for positive results. The following theorem of Bourgain [5] from 2010 (extending his own work [2] from 2003) treats the case A“B: Theorem 1.2 (Bourgain).Given αP p0,1qand γ, κ ą0, there exist ǫ0, ǫ ą0such that that the following holds for δą0sufficiently small. Let νbe a probability measure on r0,1ssatisfying νpBpx, rqq ď rγfor all xPRand 0ă rďδǫ0. Let additionally AĂ r0,1sbe a δ-separated set with |A| ě δ´α, which also satisfies the non-concentration condition |AXBpx, rq| ď rκ|A|for xPRand δďrďδǫ0. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 3 Then, there exists a point cPsptpνqsuch that |A`cA|δěδ´α´ǫ.(1.3) Remark 1.4.Bourgain’s theorem admits the following stronger version, which, to the best of my knowledge, was first stated and proved by He [17, Theorem 1] (see also [5, (7.43), p. 221] for a slightly weaker result): under the assumptions of Theorem 1.2, there exists a point cPsptpνqsuch that |πcpGq|δěδ´α´ǫfor all subsets GĂAˆAof cardinality |G| ě δǫ|A|2. Here πcpx, yq “ x`cy. This version is useful for proving lower bounds for dimHpA`cAq. Bourgain also proved such lower bounds in [5, Theorem 4] without explicitly mentioning the stronger version of Theorem 1.2: while his proof is correct, it requires some care from the reader to extract all the details. Applying the stronger version directly is simpler, see [17, Theorem 2]. To see the connection between Theorem 1.2 and the ABC problem, let CĂ r0,1sbe aδ-separated set satisfying |CXBpx, rq| ď rγ|C|for all xPRand δďrďδǫ0. Then the uniformly distributed probability measure νon the δ-neighbourhood of Csatisfies νpBpx, rqq .rγ, and it follows from (1.3) that there exists cPCwith |A`cA|δěδ´α´ǫ. Theorem 1.2 formally only treats the case A“B, but an inspection of its proof (or, more directly, an application of [5, Theorem 3]), reveals that the result remains valid for two different δ-separated sets A, B Ă r0,1s, provided that |A| “ |B|, or at least |B| « |A|. The precise meaning of "«" is defined via the various constants appearing in [5, Theorem 3]. To the best of my knowledge, Theorem 1.2 does not cover the case where |A| “ δ´α and |B| “ δ´βwith βăα(the case βąαis not relevant here: then |A`cB|δ&|B| " |A| for any cPRwith |c| „ 1). The following conjecture would correspond to the assumption |B||C| " |A|which suffices in the discrete variants (on Rand Zp) of the ABC sum-product problem: Conjecture 1.5. Let α, β, γ P p0,1qwith βďαand γąα´β. Assume that A, B, C Ă r0,1s are δ-separated sets with cardinalities |A| ď δ´α,|B| “ δ´β, and |C| “ δ´γ. Assume moreover that |BXBpx, rq| .rβ|B|and |CXBpx, rq| .rγ|C|for all xPRand rą0. Then, there exists ǫ“ǫpα, β, γq ą 0and a point cPCsuch that |A`cB|δ&α,β,γ δ´ǫ|A|. The lower bound for γin Conjecture 1.5 is necessary, but the non-concentration assumptions on Band Care quite likely not sharp. The main result of this paper is the following partial result, where the lower bound γąα´βis upgraded to γą pα´βq{p1´βq: Theorem 1.6. Let 0ăβďαă1and κą0. Then, for every γP ppα´βq{p1´βq,1s, there exist ǫ0, ǫ, δ0P p0,1 2s, depending only on α, β, γ, κ, such that the following holds. Let δP2´N with δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1ssatisfy the following hypotheses: (A) |A| ď δ´α. (B) |B| ě δ´β, and Bsatisfies the following Frostman condition: |BXBpx, rq| ď rκ|B|, δ ďrďδǫ0. Further, let νbe a Borel probability measure with sptpνq Ă r1 2,1s, and satisfying the Frostman condition νpBpx, rqq ď rγfor xPRand 0ărďδǫ0. Then, there exists a point cPsptpνqsuch that the following holds: if GĂAˆBis any subset with |G| ě δǫ|A||B|, then |πcpGq|δěδ´ǫ|A|,where πcpx, yq “ x`cy. 4 TUOMAS ORPONEN Theorem 1.6 with α“βrecovers Theorem 1.2, and the stronger version in Remark 1.4. In fact, Theorem 1.6 is formally stronger than Theorem 1.2, since Theorem 1.6 does not impose any non-concentration conditions on A. This is useful in proving Corollary 1.11 below. Theorem 1.6 easily yields the following corollary for Hausdorff dimension: Corollary 1.7. Let 0ăβďαă1and κą0. Then, there exists η“ηpα, β, κq ą 0such that if A, B ĂRare Borel sets with dimHA“α,dimHB“β, then dimHtcPR: dimHpA`cBq ď α`ηu ď α´β 1´β`κ. The case α“βis already contained in Bourgain’s paper [5]. The reduction from Theorem 1.6 to Theorem 1.7 is a standard pigeonholing argument, and goes the same way as the proof of [17, Theorem 2]. For completeness, I give the details in Section 5.6. A "continuous" version of Conjecture 1.5 would imply that the number pα´βq{p1´βqin Corollary 1.7 can be replaced by α´β. The lower bound on |πcpGq|δin Theorem 1.6 is indispensable for deducing Corollary 1.7, but makes Theorem 1.6 difficult to prove with a direct assault. Instead, Theorem 1.6 will be formally reduced to the following simpler version, which only treats G“AˆB: Theorem 1.8. Let 0ăβďαă1and κą0. Then, for every γP ppα´βq{p1´βq,1s, there exist ǫ, ǫ0, δ0P p0,1 2s, depending only on α, β, γ, κ, such that the following holds. Let δP2´N with δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1ssatisfy the following hypotheses: (A) |A| ď δ´α. (B) |B| ě δ´β, and Bsatisfies the following Frostman condition: |BXBpx, rq| ď rκ|B|, δ ďrďδǫ0. Further, let νbe a Borel probability measure with sptpνq Ă r0,1s, satisfying the Frostman condition νpBpx, rqq ď rγfor xPRand δďrďδǫ0. Then, there exists cPsptpνqsuch that |A`cB|δěδ´ǫ|A|. Theorem 1.8 is the heart of the paper, but as far as I know, it is also news that Theorem 1.6 can be literally reduced to Theorem 1.8. This takes some work, but is mostly a matter of "standard techniques" in additive combinatorics. Since these details can be carried out without reference to the rest of the paper, they are postponed to Section 5.1. Remark 1.9.As written above, Theorem 1.6 is deduced from Theorem 1.8 in Section 5.1. A variant of this problem is the following. Assume that we want to prove Theorem 1.6 with a fixed non-concentration exponent "κ". Can we deduce it from the version of Theorem 1.8 with the same κ? The answer is "almost": it turns out that in order to deduce Theorem 1.6 for a fixed non-concentration exponent κą0, we only need to invoke Theorem 1.6 with non-concentration exponent ¯κP p0, κqarbitrarily close to κ: however, the values of the constants ǫ, δ0produced by the argument will tend to 0as ¯κÕκ. The reductions in Section 5.1 will be written in such a way that this claim becomes apparent – and the matter will be further refreshed in Remarks 5.11,5.37, and 5.57. 1.1. Related work. A relevant piece of recent literature is the paper of Guth, Katz, and Zahl [13], where the authors extend an argument (due to Garaev [11]) from finite fields to give a new, relatively simple, proof of Bourgain’s Theorem 1.2. Given that the Zp analogue of Conjecture 1.5 is known [29], it may be plausible that Conjecture 1.5 can be solved by extending the Zpargument in the fashion of Guth, Katz, and Zahl. I was not ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 5 able to carry this out, and here is why. The proof in [29] is chiefly based on the following lemma: if A, B ĂZpare sets with |A| “ pαand |B| “ pβ, then for every ηą0there exists an integer n“npα, β, ηq P N, and choices a1,...,anP ˘Asuch that |a1B`...`anB|&p´ηmint|A||B|, pu.(1.10) I was not able to extend the finite field techniques in [29] to (directly) prove a δ-discretised analogue of (1.10). However, once Theorem 1.8 is known, it can be applied to make partial progress towards a δ-discretised analogue of (1.10) (a sharper result would follow from Conjecture 1.5 in the same way): Corollary 1.11. Let β, γ P p0,1qand 0ăηăγp1´βq. Then, there exists ǫ0, δ0ą0and nPN, depending on β, γ, η, such that the following holds for all δP p0, δ0s. Let B, C Ă pδ¨ZqXr0,1s be non-empty sets satisfying |BXBpx, rq| ď rβ|B|and |CXBpx, rq| ď rγ|C|(1.12) for xPRand δďrďδǫ0. Then, there exist points c1,...,cnPCsuch that |c1B`...`cnB|δěδ´γ´βp1´γq`η“δ´β´γp1´βq`η. Remark 1.13.A classical projection theorem of Kaufman [20] implies the existence of cPC such that |B`cB|δ'maxtδ´β, δ´γu. For γąβ, a recent sharpening of Kaufman’s theorem by the author and Shmerkin [28] even yields |B`cB|δěδ´γ´ηfor some η“ ηpβ, γq ą 0, and for δą0small enough (to be clear, this statement is only a corollary of the main result in [28]). In comparison, Corollary 1.11 gives a far more substantial improvement, but at the cost of adding the number of summands. I give the simple proof straight away. Proof of Corollary 1.11.Start by applying Theorem 1.8 with α:“β`γp1´βq´ηP pβ, 1q, β, κ :“β, and γ. Note that γą pα´βq{p1´βq, so the parameters are admissible. Let ǫ, ǫ0, δ0P p0,1 2sbe the constants given by Theorem 1.8 with α, β, γ, κ. Let ν:“ |C|´1¨H0|Cbe the normalised counting measure on C, which satisfies the Frostman condition νpBpx, rqq ď rγfor all xPRand δďrďδǫ0by (1.12). We also note that |B| ě δ´βby (1.12) applied with r“δ, and Bsatisfies the κ“β-dimensional Frostman condition required in Theorem 1.8. We construct a sequence of sets HnĂδ¨Z,nPN, with the following greedy algorithm. We first define H1:“ pc1Bqδarbitrarily, where Aδ:“ pδ¨ZqXApδq. Then, we assume that Hnhas already been defined for some ně1, and we let Hn`1:“Hn`pcn`1BqδĂδ¨Z, where cn`1PCmaximises |Hn`cB|δamong all choices cPC. We observe (by induction) that HnĂ pδ¨ZqXr0, ns, so |Hn| ď nδ´1. For arbitrary NPNwith Ně2, it follows from the pigeonhole principle that there exists nP t1,...,N ´1usuch that |Hn`1| ď 2pNδ´1q1{pN´1q|Hn| ď 4δ´1{pN´1q|Hn|.(1.14) Indeed, if the first inequality failed for every nP t1,...,N ´1u, then |HN| ą 2pNδ´1q1{pN´1q|HN´1| ą ...ą2N´1pNδ´1qpN´1q{pN´1q|H1| ě 2N´1Nδ´1, 6 TUOMAS ORPONEN contradicting that HNĂ pδ¨ZqXr0, Ns. By definition of Hn`1, (1.14) implies |Hn`cB|δ.|Hn`1| ď 4δ´1{pN´1q|Hn|, c PC. (1.15) We now choose NPNso large that 4δ´1{pN´1qďδ´ǫ{2, where ǫ“ǫpα, β, γq ą 0was one of the constants produced by Theorem 1.8. Since Band νsatisfy the hypotheses of Theorem 1.8, we see from (1.15) that A:“Hnmust fail the hypotheses. However, the only hypotheses on Ain Theorem 1.8 are AĂ pδ¨ZqXr0,1sand |A| ď δ´α. Of course HnĆ r0,1s, but this is not really relevant: we may find kP t0,...,n´1usuch that |HnXrk, k `1s| ě 1 N|Hn|. Now, defining instead A:“ pHnXrk, k `1sq ´tku, we have AĂ pδ¨ZqXr0,1s, and |A`cB|δďδ´ǫ{2|Hn| ď δ´ǫ|A|by (1.15), for δą0so small that δ´ǫ{2ěN. This violates Theorem 1.8, unless |Hn| ě |A| ą δ´α“δ´β´γp1´βq`η, and this is what the corollary claimed.  The ABC sum-product problem is, of course, related to the highly active area of sumproduct theory. The main open question is the Erd˝os-Szemerédi sum-product conjecture [8]: if AĂRor AĂZpis a finite set (pPNis prime), the E-S conjecture asks to prove that maxt|A`A|,|A¨A|u &ǫ|A|2´ǫ, ǫ ą0. The research around this problem is too active to survey here: I only mention the papers [32] of Rudnev-Stevens and [24] of Mohammadi-Stevens for some current world records, and further references. For results on the the δ-discretised variant of the Erd˝os-Szemerédi problem, see [13] by Guth-Katz-Zahl, and [7] by D ˛abrowski, the author, and Villa. Bourgain’s δ-discretised sum-product estimate, Theorem 1.2, has been extended in various ways beyond the real line. For example, He [17] found a version of the theorem in Rn. Closely related are also the works [3,6] by Bourgain-Gamburd, [16] by He, [18] by He-de Saxcé, [1] by Benoist-de Saxcé, and [21] by Li. These papers contain δ-discretised sum-product or product theorems in various Lie groups. Viewing the δ-discretised ABC sum-product problem as a special case of a δ-discretised incidence problem between points and δ-tubes in R2, the papers [10,14] are also relevant. Theorems 1.2 and 1.8 can be viewed as statements concerning linear projections of planar sets, as discussed more in the next subsection. Starting with this interpretation, one may ask if analogous statements hold for non-linear projections. Examples of particular interest are the pinned distance projections △xpyq “ |x´y|and the radial projections πxpyq “ px´yq{|x´y|. Again, the literature is too broad for a survey, but see the recent papers [36] by Shmerkin, [37] by Shmerkin-Wang, and [31] by Raz-Zahl for recent exciting developments and more references. Finally, Conjecture 1.5 was recently solved by the author [27] for Ahlfors-regular sets A, B Ă r0,1s. In fact, a much stronger result can be obtained for such sets. Let α, β P p0,1q. Assume that A, B ĂRare closed sets, Ais α-Ahlfors-regular and Bis β-Ahlforsregular. Then dimHtcPR: dimpA`cBq ă α`ηu “ 0 for η:“βp1´αq{p2´αq ą 0. (The paper [27] also contains a δ-discretised version.) ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 7 1.2. Comparison to classical projection theorems. A popular topic in fractal geometry is to study the orthogonal projections of subsets of Rd. In this section we will see what "classical" projection theorems in fractal geometry have to say about the size of A`cB. For ePS1, let πe:R2Ñspanpeqbe the orthogonal projection. A theorem of Kaufman [20] from 1968, sharpening a seminal result of Marstrand [22], states the following: if KĂR2is a compact set with dimension dimHK“t, then ΣpK, sq:“dimHtePS1: dimHπepKq ď su ď s, 0ďsăt. (1.16) Another classical estimate, due Peres-Schlag [30] but building on a Fourier-analytic technique introduced by Falconer [9], shows that ΣpK, sq ď maxt1`s´t, 0u,0ďsďt. (1.17) A folklore conjecture (made explicit in [25]) proposesto improve (1.16)-(1.17) to ΣpK, sq ď maxt2s´t, 0ufor 0ďsăt. Bourgain [5] showed that ΣpK, sq Ñ 0as sÑt{2, which supports the conjecture. A recent preprint [28] of the author and Shmerkin additionally shows that ΣpK, sq ď s´ǫfor some ǫ“ǫps, tq ą 0, for all 0ďsăt. The connection between orthogonal projections and the A`cB problem is the following. Take K“AˆB, where A, B ĂR. Then, for ePS1ztp0,1q,p0,´1qu, the projection πepKqcan, up to rescaling, be rewritten as A`cB, for a suitable c“cpeq P R. With this in mind, the bounds (1.16)-(1.17) can be used to deduce the following. Let 0ăβďαă1. Assume that A, B ĂRare Borel sets with dimHA“αand dimHB“β. Then, (1.16)-(1.17) applied with t:“dimHpAˆBq ě α`βyield dimHtcPR: dimHpA`cBq ď αu ď mintα, 1´βu. In contrast, letting ηÑ0in Corollary 1.7 gives the upper bound pα´βq{p1´βq. This bound is ăαfor all αă1, and also ă1´βwhenever 0ăβďαă3 4. If αą3 4, then the "1´β" estimate coming from (1.17) is better for some values of β, e.g. β“1 2. The conjectured bound ΣpK, sq ď maxt2s´t, 0uwould imply the (Hausdorff dimension version of) Conjecture 1.5: dimHtcPR: dimHpA`cBq ď αu “ ΣpAˆB, αq ď maxt2α´t, 0u ď α´β. To summarise, Corollary 1.7 is stronger than all previous results in the case K“AˆB and s“α“dimHAă3 4, whereas the conjecture ΣpK, sq ď maxt2s´t, 0uis even stronger than (the Hausdorff dimension version of) Conjecture 1.5. 1.3. Paper outline and proof sketch. The proof of Theorem 1.6 has two distinct components: the first one is a reduction to Theorem 3.28, which differs from Theorem 1.6 in the following aspects: (a) νsatisfies a Frostman condition on all scales δďrď1, (b) the set Bhas small doubling, that is |B`B| ď δ´ǫ|B|, and (c) the conclusion |πcpGq|δěδ´ǫ|A| is only required for G“AˆB. These reductions are performed in several steps: Theorem 3.28 §3.3 ùñ Theorem 3.15 §3.2 ùñ Theorem 3.1 §3.1 ùñ Theorem 1.8 §5.4 ùñ Theorem 5.4 §5.3 ùñ Theorem 5.3 §5.5 ùñ Theorem 1.6. 8 TUOMAS ORPONEN The outline of the paper is that the reduction from Theorem 1.8 to Theorem 3.28 is performed first, then Theorem 3.28 is proved with a direct argument, and finally Theorem 1.6 is reduced to Theorem 1.8 in Section 5.1. The additional assumptions (a)-(c) in Theorem 3.28 are technically important. However, at the current level of discussion, all the theorems above are indistinguishable. So, for example, the reader may think that the following outline concerns the proof of Theorem 1.8, which has the simplest statement. For the sake of exposition, I make the following additional assumptions on Aand B. Both sets have a "tree" (or "Cantor set") structure: for a suitable parameter mPN, each dyadic interval IPDms intersecting Acontains exactly RApsqsub-intervals in Dmps`1q which intersect A. The same is assumed of B. The numbers RApsqand RBpsqare known as the branching numbers of Aand B, respectively. Assume that the scale parameter δą0 has the special form δ“2´mN for some NPN(thus RApsq “ 1“RBpsqfor sěN, since A, B were assumed to be δ-separated). We make even more assumptions: (P1) For every sPN, either RBpsq “ 1or RApsq “ 2m. (P2) |B| “ δ´β, and for every sPN, either RBpsq “ 1or RBpsq “ 2m. Property (P2) needs the small doubling assumption |B`B| ď δ´ǫ|B|. Now, as we will see in a moment, the key question turns out to be: given a scale sPNwith RBpsq “ 1, what upper bound can we guarantee for RApsq? It turns out that we can easily use (P1)-(P2) to deduce an answer. Assume that RApsq ě 2Γmfor all sP t0,...,N ´1u “:rNswith RBpsq “ 1. Write N:“ tsP rNs:RBpsq “ 1u, and note that RApsq “ 2mfor all sP rNszNby assumption (P1). Now, we may calculate a lower bound on the cardinality of Aas follows: 2αmN ě |A| “ ź sPrNs RApsq “ ź sPrNs z N RApsq¨ ź sPN RApsq ě 2mpN´|N|q ¨2Γm|N|.(1.18) On the other hand, by assumption (P2), we have 2βmN “ |B| “ ź sPrNs z N 2m“2mpN´|N|q,(1.19) so may solve N´|N| “ βN and |N| “ p1´βqN. Plugging this information into (1.18) yields Γď pα´βq{p1´βq. This is where the numerology in Theorem 1.8 comes from. Namely, the argument above shows that if Γą pα´βq{p1´βq, then there exists at least one scale sP rNssuch that RApsq ď 2Γm. In fact, the same must be true for a positive fraction of the scales, say GĂ rNs, where |G|{Nonly depends on Γ´pα´βq{p1´βq. After this observation, we focus attention separately on pieces of AˆBof the form pAXIqˆpBXJq, where I, J PDms are intervals intersecting A, B, respectively, and sPG. By definition, |AXI|mps`1q“RApsq ď 2Γmfor some Γslightly larger than pα´βq{p1´βq. To be precise, we choose pα´βq{p1´βq ă Γăγ, where γis the Frostman exponent of the measure νin Theorem 1.8. If we additionally knew that |BXJ|2´mps`1q“RBpsq ě 2ǫm for some ǫą0, and the points in BXJare well enough separated, we could at this point use an elementary argument (essentially the "potential theoretic method" due to Kaufman [20]) to deduce that |pAXIq`cpBXJq|2´mps`1qě2ǫm|AXI|2´mps`1q(1.20) ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 15 With these choices of constants, fix δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1sbe sets, and let νbe a Borel probability measure on r0,1s, satisfying the hypotheses in Theorem 3.1. To land in a situation where Theorem 3.15 becomes applicable, we consider initially the measure ¯ν:“ν˚p´νq, where ´νpAq:“νp´Aq. Evidently sptp¯νq Ă r´1,1s. As Bourgain shows in [5, (5.5)], the measure ¯νhas the property ¯νpBpx, rqq ď 4¨¯νpBp0, rqq ď 4¨sup yPR νpBpy, rqq, x PR, r ą0.(3.20) Now, let c0ě0be the infimum of the numbers such that ¯νpBp0, c0qq ą 5¨cγ 0,(3.21) if any such numbers exist. Evidently c0P r0,1s, since ¯νis a probability measure on Bp0,1q. If no c0as in (3.21) exists, then let c0:“maxt|c|:cPsptp¯νqu, and note that 5¨cγ 0ě¯νpBp0, c0qq ě ¯νpBp0,1qq “ 1ùñ c0ě5´1{γěδǫ0,(3.22) assuming here that δ0ěδis sufficiently small in terms of γ, ǫ0. Assume then that c0, as in (3.21), exists. Since supyPRνpBpy, rqq ď rγfor all 0ărďδǫ0by assumption, (3.20) implies that c0ěδǫ0. In both cases, c0ěδǫ0. Moreover, we note that sptpνqXtc0,´c0u ‰ Hin both cases (in the non-trivial case, otherwise some smaller value of c0would also satisfy (3.21)). We then consider the re-normalised measure ¯νc0defined by ¯νc0pHq:“1 ¯νpBp0,c0qq ¨¯ν|Bp0,c0qpc0¨Hq, H ĂR, which satisfies sptp¯νc0q “ c´1 0¨pspt ¯νX¯ Bp0, c0qq Ă r´1,1s. Clearly ¯νc0is a Borel probability measure. Moreover, if xPRand rP rδ, 1s, then, assuming that c0P r0,1swas defined via (3.21), we have ¯νc0pBpx, rqq ď ¯νpBpc0x, c0rqq ¯νpBp0, c0qq (3.20) ď4¨¯νpBp0, c0rqq 5¨cγ 0ď4¨5¨pc0rqγ 5¨cγ 0“4¨rγ. If c0was, instead, defined as c0“maxt|c|:cPsptp¯νqu, then ¯νpBp0, c0qq “ 1, so ¯νc0pBpx, rqq ď ¯νpBpc0x, c0rqq ¯νpBp0, c0qq (3.20) ď4¨¯νpBp0, c0rqq ď 20 ¨pc0rqγď20 ¨rγ. The same estimates are also true for rą1, since }¯νc0} “ 1. Therefore, in any case ¯νc0 satisfies the hypotheses of Theorem 3.15 with Frostman constant 20. We will not apply Theorem 3.15 directly to the sets A, B, but rather to A, pc0Bqδ, where pc0Bqδ“ pδ¨ZqXpc0Bqpδq Ă pδ¨ZqXr0,1s. Evidently |pc0Bqδ|&c0|B| ě δǫ0|B|by (3.22). It follows that |pc0Bqδ`pc0Bqδ|.|B`B| ď δ´ǫB|B|.δ´ǫ0´ǫB|pc0Bqδ|. Since ǫ0`ǫBď¯ǫB{2by (3.18), and if δą0is sufficiently small, we conclude that pc0Bqδ satisfies the small doubling assumption in Theorem 3.15 with constant ¯ǫB. We moreover claim that pc0Bqδsatisfies the Frostman condition |pc0BqδXBpx, rq| ď rκ{2|pc0Bqδ|for all 16 TUOMAS ORPONEN δďrďδ¯ǫ0. To see this, fix δďrďδ¯ǫ0ďδ2ǫ0ďc0δǫ0(by (3.18) and (3.22)), and note that |pc0BqδXBpx, rq| .|BXBpx, c´1 0rq| ď pc´1 0rqκ¨|B| .c´2 0¨rκ¨|pc0Bqδ| ďδ´2ǫ0¨rκ{2¨rκ{2¨|pc0Bqδ| (3.18) ďδ2ǫ0¨rκ{2¨|pc0Bqδ|. This implies |pc0BqδXBpx, rq| ď rκ{2|pc0Bqδ|, provided that δ0ěδis sufficiently small. We have now shown that Theorem 3.15 is applicable with the parameters α, β, γ, κ{2to the the sets A, pc0Bqδ, and the measure ¯νc0. Since δďδ0ď¯ δ0, Theorem 3.15 implies the existence of a point cPsptp¯νc0q Ă r´1,1sX c´1 0¨psptpνq´sptpνqqsuch that |A1`cpc0Bqδ| ě δ´¯ǫ|A|(3.23) for all subsets A1ĂAwith |A1| ě p1´¯ρq|A|. Note that the point cPsptp¯νc0qin (3.23) can be written as c“c´1 0¨pc1´c2qfor certain points c1, c2Psptpνq. Therefore |A1`pc1´c2qB|δ“ |A1`c1´c2 c0¨c0B|δ&|A1`cpc0Bqδ|δěδ´¯ǫ|A|(3.24) for all A1ĂAwith |A1| ě p1´¯ρq|A|. We now claim that there exists ¯cP tc1, c2usuch that |A1`¯cB|δěδ´ǫ|A|, A1ĂA, |A1| ě p1´ρq|A|,(3.25) assuming that δ0ěδis small enough, depending on ǫ, ρ. This will prove Theorem 3.1. Assume that (3.25) fails for both ¯cP tc1, c2u, and let A1 1, A1 2ĂAbe subsets of cardinalities |A1 j| ě p1´ρq,jP t1,2u, such that |A1 1`c1B| ă δ´ǫ|A|and |A1 2`c2B| ă δ´ǫ|A|.(3.26) We first observe from the second inequality in (3.26) that |A1 2`c2B|δďδ´ǫ|A| ď 2δ´ǫ|A1 2|. By Lemma 3.16, for Ně1there exists a subset A2 2ĂA1 2of cardinality |A2 2| ě p1´ρq|A1 2| such that |A2 2´c2B|δ.ρ,N pδ´ǫq2N|A1 2|1`1{N.δ´ǫ¨2N`2´1{N|A2 2|.(3.27) We apply this with N„log2p1{ǫqsatisfying ǫ¨2N`2„?ǫ. Since with this choice ǫ¨2N`2„ ?ǫ!1{log2p1{ǫq „ 1{N, we have ǫ¨2N`1`1{Nď2{N„1{log2p1{ǫq, and we deduce from (3.27) that |A2 2´c2B|δ.ǫ,ρ δ´C0{log2p1{ǫq|A2 2| for some absolute constant C0ą0. Now, recall that |A1 1| ě p1´ρq|A|and |A2 2| ě p1´ρq|A1 2| ě p1´?ρq|A|. Consequently, the intersection A1:“A1 1XA2 2satisfies |A1| ě p1´ρ´?ρq|A|. Evidently, |A1`c1B| ď δ´ǫ|A|.δ´C0{log2p1{ǫq|A1|and |A1´c2B|.ρ,ǫ δ´C0{log2p1{ǫq|A1|. By Lemma 3.3, there exists a further subset A2ĂA1with |A2| ě p1´ρq|A1| ě p1´ρqp1´ρ´?ρq|A|(3.19) ě p1´¯ρq|A| ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 17 such that |A2`c1B´c2B|δ.ǫ,ρ δ´2C0{log2p1{ǫq|A2|(3.17) ďδ´¯ǫ{2|A|. This contradicts (3.24) for δą0small enough, depending on ǫ, ρ, and proves (3.25). The proof of Theorem 3.1 is complete.  3.3. Removing reference to subsets. In the previous reductions, we have upgraded the assumptions of Theorem 1.8 in two ways: we have arranged the set Bto have small doubling, and the Frostman constant of νto be 20. However, there has been a price: whereas Theorem 1.8 only claims that |A`cB|δěδ´ǫ|A|for some cPsptpνq, Theorem 3.15 claims the existence of cPsptpνqsuch that |A1`cB|δěδ´ǫ|A1|for all A1ĂA with |A1| ě p1´ρq|A|. It turns out that this innocent-looking difference makes Theorem 3.15 difficult to prove with a direct assault. Therefore, we need a final reduction to the following statement: Theorem 3.28. Let 0ăβďαă1and κą0. Then, for every γP ppα´βq{p1´βq,1s, there exist ǫ, ǫ0, ǫB, δ0P p0,1 2s, depending only on α, β, γ, κ, such that the following holds. Let δP2´Nwith δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1ssatisfy the following hypotheses: (A) |A| ď δ´α. (B) |B| ě δ´β, and Bsatisfies the following Frostman condition: |BXBpx, rq| ď rκ|B|, δ ďrďδǫ0. Assume moreover that |B`B| ď δ´ǫB|B|. Further, let νbe a Borel probability measure with sptpνq Ă r´1,1ssatisfying the Frostman condition νpBpx, rqq ď 40 ¨rγfor xPRand rěδ. Then, there exists a point cPsptpνqsuch that |A`cB|δěδ´ǫ|A|. Theorem 3.28 only differs from Theorem 3.15 in its (superficially) weaker conclusion, and in that the Frostman constant of νhas increased from 20 to 40. Proof of Theorem 3.15 assuming Theorem 3.28.Fix the parameters 0ăβďαă1,κ, and γą pα´βq{p1´βqfrom Theorem 3.15. As usual, our task is to find the parameters ǫ, ǫ0, ǫB, δ0, ρ such that Theorem 3.15 is satisfied. In doing so, we apply Theorem 3.28 to the parameters 0ăβď¯αă1and κ, where ¯αP pα, 1qis arbitrary such that the key inequality γą p¯α´βq{p1´βqstill holds. Then, we let ¯ǫ, ¯ǫ0,¯ǫB,¯ δ0ą0(3.29) be the constants given by Theorem 3.28 with parameters ¯α, β, κ, γ. We now begin defining the parameters ǫ, ǫ0, ǫB, δ0, ρ. We set ǫ0:“¯ǫ0and ǫB“¯ǫB.(3.30) We will need that δ0ď¯ δ0, and there will be an additional (simple) dependences on the allowed parameters, which will be explained when they arise. To define the parameters ǫ, ρ, fix a natural number N„1{¯ǫ, so that the following holds: pN´1q´1ă¯ǫ{2.(3.31) Then, let ǫ:“¯α´α 2N`1.(3.32) 18 TUOMAS ORPONEN Finally, define ρą0, depending only on ¯ǫ, so small that `2p1´p1´ρqNq˘1{2N ď1 2.(3.33) This is possible, since the inequality is clearly true for ρ“0. We now make the counter assumption that Theorem 3.15 fails for certain δP p0, δ0s, A, B Ă pδ¨Zq X r0,1s, and a Borel probability measure νon r´1,1s, satisfying the hypotheses of Theorem 3.15 with parameters α, β, κ, γ, and the constants ǫ0, ǫ, δ0described above. This means that for every cPC“sptpνq, there exists a subset AcĂAwith the properties |Ac| ě p1´ρq|A|and |Ac`cB|δďδ´ǫ|A|.(3.34) The plan is to use this information to construct a new set ¯ AĂ pδ¨ZqX r0,1s, and a new probability measure ¯νon r´1,1s, such that the triple ¯ A, B, ¯νsatisfies the hypotheses of Theorem 3.28 with parameters ¯α, β, κ, γ and constants ¯ǫ0,¯ǫB, but nevertheless |¯ A`cB| ă δ´¯ǫ|¯ A|for all cPsptp¯νq. This contradiction will complete the proof of Theorem 3.15. Given such a set AcĂAfor every cPC, we observe that ż...ż|Ac1X...XAcN|dνpc1q¨¨¨dνpcNq ě p1´ρqN|A|(3.35) by Hölder’s inequality. Consider the set Ω :“ tpc1,...,cNq P CN:|Ac1X...XAcN| ě 1 2|A|u. If "I" temporarily stands for the integral in (3.35), we have p1´ρqN|A| ď IďνNpΩcq¨ 1 2|A|`p1´νNpΩcqq¨|A|, which can be rearranged to νNpΩcq ď 2p1´p1´ρqNq. Consequently νNpΩq ě 1´2p1´p1´ρqNq “: 1 ´θ0.(3.36) For c1,...,cnPCfixed, we define Ωc1¨¨¨cn:“ tpcn`1,...,cNq P CN´n:pc1,...,cNq P Ωu. It follows from Fubini’s theorem that νN´npΩc1¨¨¨cnq “ żνN´n´1pΩc1¨¨¨cncqdνpcq(3.37) for all c1,...,cnPC, and 1ďnďN´2. The same remains true for n“0, if the left hand side is interpreted as νNpΩq. Equation (3.37) also remains valid for n“N´1if we define the notation νN´n´1“ν0as follows: ν0pΩc1¨¨¨cN´1cq:“1Ωpc1,...,cN´1, cq.(3.38) We will use this notation in the sequel. For pc1,...,cNq P Ωfixed, we write Ac1¨¨¨cN:“Ac1X...XAcNùñ |Ac1¨¨¨cN| ě 1 2|A|.(3.39) We now construct a sequence of sets HnĂδ¨Z,1ďnďN. At the same time, we will construct subsets C1,...,CNĂC, and points cnPCn,1ďnďN, with the properties νN´npΩc1¨¨¨cnq ě 1´θnand νpCnq ě 1´θn,1ďnďN, (3.40) where we define inductively θn:“aθn´1ěθn´1. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 19 In particular, the first part of (3.40) with n“Nshows that pc1,...,cNq P Ω, recall the notation (3.38). As a second remark, recalling the definition of θ0“2p1´p1´ρqNq, and combining this with the definition of ρin (3.33), one sees that θnď1 2for all 1ďnďN. To begin with, we define C1:“ tcPC:νN´1pΩcq ě 1´θ1u, and we choose an arbitrary element c1PC1. Since 1´θ0ďνNpΩq “ żνN´1pΩcqdνpcq ď νpCc 1q¨p1´θ1q`p1´νpCc 1qq “ ´θ1¨νpCc 1q`1 by (3.36), we observe that νpCc 1q ď θ0{θ1“θ1, and consequently νpC1q ě 1´θ1. In particular C1‰ H. We then define H1:“ pc1Bqδ. Assume inductively that H1,...,Hnand C1,...,CnĂC, and cjPCj,1ďjďnďN´1, have already been constructed, and satisfy (3.40). We pick an element cn`1PCn`1, where Cn`1:“ tcPC:νN´n´1pΩc1¨¨¨cncq ě 1´θn`1u,1ďnďN´1. For n“N´1, the notation νN´n´1pΩc1¨¨¨cncqshould be interpreted as in (3.38), so CN“ tcPC:1Ωpc1,...,cN´1, cq ě 1´θNu “ tcPC:pc1,...,cN´1, cq P Ωu. For an arbitrary choice cn`1PCn`1, we note that the first part of (3.40) is satisfied with index "n`1", by the definition of Cn`1. The set Cn`1also satisfies the second part of (3.40) with index "n`1", since 1´θn (3.40) ďνN´npΩc1¨¨¨cnq(3.37) “żνN´n´1pΩc1¨¨¨cncqdνpcq ď ´θn`1¨νpCc n`1q`1, and consequently νpCc n`1q ď θn{θn`1“θn`1, and νpCn`1q ě 1´θn`1. Whereas c1PC1was chosen arbitrarily, the element cn`1PCn`1is chosen in such a way that the quantity |Hn`cn`1B|δis maximised, among all possible choices cn`1P Cn`1. We then define Hn`1:“Hn`pcn`1Bqδ. Continuing in this manner produces a distinguished sequence pc1,...,cNq P Ω, which we fix for the remainder of the argument, and a sequence of sets H1,...,HN. Note that HnĂ pδ¨Zq X r0, Nsfor all 1ďnďNby a straightforward induction, so |Hn| ď 2Nδ´1. Therefore, by the pigeonhole principle, there exists an index nP t1,...,N ´1usuch that |Hn`1| ď p2Nδ´1q1{pN´1q|Hn| ď 4δ´1{pN´1q|Hn|.(3.41) For this particular index nP t1,...,N ´1u, we then have |Hn`cB|δ.|Hn`1| ď 4δ´1{pN´1q|Hn|for all cPCn`1by the definition of Hn`1, and therefore |Hn`cB|δďδ´¯ǫ{2|Hn|, c PCn`1,(3.42) recalling (3.31), and assuming that δą0is small enough. We now claim that (3.42) violates Theorem 3.28 with parameters ¯α, β, κ, γ, and with the objects ¯ A:“Hn, B, and ¯ν:“νpCn`1q´1¨ν|Cn`1.(3.43) We need to check the following items to contradict Theorem 3.28: 20 TUOMAS ORPONEN (a) |¯ A| ď δ´¯α, (b) |B| ě δ´βand |B`B| ď δ´¯ǫB|B|, and Bsatisfies a Frostman condition with exponents κand ¯ǫ0, (c) ¯νsatisfies a Frostman condition with exponent γand constant 40. Point (b) is true by assumption (and since we chose ǫB“¯ǫBand ǫ0“¯ǫ0in (3.30)), so only (a) and (c) need to be verified. We first use the Plünnecke-Ruzsa inequality to establish (a), assuming that δą0is sufficiently small in terms of N, ¯α. Clearly ¯ Acan be written as a sum of nďNsets of the form pcmBqδ, for some 1ďmďn, where cmis an index in the (fixed) sequence pc1,...,cNq P Ω. Noting that Ac1¨¨¨cNĂAcmĂA, each of these sets individually satisfies |Ac1¨¨¨cN`pcmBqδ|.|Acm`cmB|δ (3.34) ďδ´ǫ|A|(3.39) ď2δ´ǫ|Ac1¨¨¨cN|. We may therefore infer that |¯ A|.N,ρ δ´2Nǫ|A| ď δ´2Nǫ´α. from Lemma 3.3. This inequality implies |¯ A| ď δ´¯αfor small enough δą0, recalling our choice of ǫat (3.32). We move to (c). Recalling (3.43), and from (3.40) that νpCn`1q ě 1´θNě1 2, we have ¯νpBpx, rqq ď 2¨νpBpx, rqq ď 40 ¨rγ, x PR, r ěδ. We have now reached a situation which violates Theorem 3.28 for the choice of parameters ¯α, β, κ, γ: the objects ¯ A, B, ¯νsatisfy all the hypotheses (by (a)-(c)), but nevertheless we have |¯ A`cB|δďδ´¯ǫ|¯ A|for all cPCn`1, a set of full ¯νmeasure, by (3.42). This violates Theorem 3.28, since ¯ǫą0was the constant associated to ¯α, β, κ, γ. Therefore the counter assumption (3.34) is false, and the proof of Theorem 3.15 is complete. To be precise, we have ignored that ¯ AĂ r0, Nsinstead of ¯ AĂ r0,1s. This can be dealt with as in the proof of Corollary 1.11, or below (3.12). We leave this to the reader.  3.4. Bonus reduction. We have now reduced the proof of Theorem 1.8 to the proof of Theorem 3.28. For notational convenience in the future, we mention one final reduction: we may assume that 1Psptpνq. Indeed, assume that Theorem 3.28 is known under this extra assumption. Then, let A, B, ν be a general triple as in Theorem 3.28. Since νis a probability measure, sptpνq Ă r´1,1sand νpBpx, rqq ď 40 ¨rγ, the point c0P sptpνqXr´1,1swith maximal absolute value satisfies |c0| ě 40´1{γ. Consider the measure ¯νpAq:“νpc0Aq. Observe that ¯νpBpx, rqq ď 40 ¨rγand sptp¯νq “ c´1 0sptpνq. Therefore 1Psptp¯νq Ă r´1,1s, so ¯νsatisfies the extra assumption. We then apply the (assumedly known) version of Theorem 3.28 to A, pc0Bqδ,¯ν. The set pc0Bqδwill have slightly worse constants than B, in a manner depending on γonly, so the theorem needs to be applied with appropriately modified parameters. Once this has been done, we find a point c“c´1 0c1Psptp¯νq, where c1Psptpνq, such that |A`cB|δ&|A`pc1{c0q¨pc0Bqδ|δ“ |A`cpc0Bqδ|δěδ´ǫ|A|, and the proof of Theorem 3.28 (without the extra assumption) is complete. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 21 4. PROOF OF THEOREM 3.28 4.1. Preliminaries. We have now reduced the proof of Theorem 1.8 to the proof of Theorem 3.28. We fix the parameters α, β, γ, κ, with 0ăβďαă1and pα´βq{p1´βq ă γď1. We also fix sets A, B Ă pδ¨Zq X r0,1sand a Borel probability measure νwith sptpνq Ă r´1,1s, satisfying all the hypotheses of Theorem 3.28 with sufficiently small constants ǫ0, ǫBą0to be determined later. For future reference, we write |A| “:δ´¯α,0ď¯αďα. (4.1) We make a counter assumption: |A`cB|δăδ´ǫ|A|for all cPsptpνq. Since we may assume that 1Psptpνqby Section 3.4, we have the assumptions |A`B| ď δ´ǫ|A|and |B`B| ď δ´ǫB|B|.(4.2) If ǫ, ǫBą0in (4.2) are small enough, depending only on α, β, κ, γ, we will be able to find a point cPsptpνqsuch that |A`cB|δěδ´ǫ|A|. This will violate the counter assumption, and prove Theorem 3.28. The necessary values of ǫ“ǫpα, β, γ, κq ą 0and ǫB“ǫBpα, β, γ, κq ą 0in (4.2) will be fixed during the proof of Proposition 4.12. 4.2. Shmerkin’s inverse theorem. In the case A“B, Bourgain [5] used an assumption of the form (4.2) to obtain, up to passing to a subset, a special multi-scale structure inside A: informally speaking, when passing from one scale to the next, either Ahas full branching, or then no branching. Similar statements have, after Bourgain’s work, been proved by Hochman [19] and Shmerkin [34] in the case where A‰B, and where Aand Bmay have completely different sizes. This is our situation, and we will apply Shmerkin’s theorem, which we state in Theorem 4.6. Definition 4.3 (δ-sets and measures, L2-norms).Let δP2´Nbe a dyadic rational. A subset of pδ¨Zq X r0,1qis called a δ-set. A probability measure supported on a δ-set is called a δ-measure. The L2-norm of a δ-measure µis defined by }µ}L2:“˜ÿ zPδ¨Z µptzuq2¸1{2 . We will only be concerned with δ-measures of the form µ“ |A|´1H0|A, where AĂ r0,1qis a δ-set. Then }µ}L2“ |A|´1{2. Definition 4.4 (Uniform sets).Let m, N PN, and set δ:“2´mN P2´N. For AĂ r0,1qand sP t0,...,N ´1u, write ImspAq:“ tIPDms :AXI‰ Hu for the collection of dyadic intervals of side-length 2´ms (these are denoted Dms) with non-empty intersection with A. We say that Ais pm, Nq-uniform if RApsq:“ |IXA|2´mps`1q, I PImspAq, is independent of the choice of IPImspAq. We may also write that Ais pm, N, RAquniform if the branching numbers RAneed emphasising. In the definition of RApsq, is it important to remember that |H|ris, by definition, the number of dyadic r-intervals intersecting H– instead of the r-covering number. This distinction has hardly mattered earlier in the paper. As in [34], we will only consider uniform sets which are also δ-sets. It was observed by Bourgain [5] that every δ-set contains a uniform subset of "comparable" cardinality. Thus, 22 TUOMAS ORPONEN the possibility of finding uniform subsets has nothing to do, yet, with an assumption like (4.2). To explain what (4.2) implies, we introduce the following terminology: Definition 4.5 (η-polarised pair).Let m, N PN,δ“2´mN , and ηą0. A pair of pm, Nquniform sets pA, Bqis pη, m, Nq-polarised, if RBpsq ą 1ùñ RApsq ě 2p1´ηqm, s P t0,...,N ´1u. If A“B, we say that A(instead of pA, Aq) is pη, m, Nq-polarised. Note that RApsq ď 2mfor all sP t0,...,N ´1u, so RApsq ě 2p1´ηqmmeans that RApsqis nearly maximal. Bourgain [5] proved that if Ais a δ-set with |A`A| ď δ´ǫ|A|, then Acontains a uniform subset A1such that |A1| ě δη|A|, and A1is η-polarised, where η“oǫp1q. This means that either RA1psq “ 1or RA1psq ě 2p1´ηqmfor all scales "s". Versions of Bourgain’s "polarisation theorem", explained above, for two different sets were found by Hochman [19] and Shmerkin [34]. Hochman first showed that if µ, ν are probability measures on r0,1q, then the entropy inequality Hpµ˚νq ď Hpµq `ǫimplies a measure-theoretic version of the polarisation phenomenon for µ, ν. The set version, below, was established by Shmerkin [34] (with a proof very different from [19]): Theorem 4.6 (Shmerkin).Let ηą0, and let mpηq P Nbe sufficiently large, depending on η. Then, for all měmpηqthere exists ǫ“ǫpη, mq ą 0such that the following holds for all large enough NPN. Let δ“ p2´mqN, and let A, B Ă r0,1sbe δ-sets such that |A`B| ď δ´ǫ|A|. Then, there exist pm, Nq-uniform sets A1ĂAand B1ĂBsuch that |A1| ě δη|A|,|B1| ě δη|B|, and pA1, B1qis pη, m, Nq-polarised. Remark 4.7.To be accurate, Theorem 4.6 is a slight refinement of Shmerkin’s theorem: [34, Theorem 2.1] literally contains the following statement: if ηą0and m0PN, then there exists m“mpη, m0q ě m0and ǫ“ǫpη, m0q ą 0as in Theorem 4.6.However, if one inspects the proof of [34, Theorem 2.1], one observes that the only dependence of m“mpη, m0q on m0is "měm0", and any choice of měm0works, provided that mis also sufficiently large in terms of η. This is precisely what Theorem 4.6 says. As another remark, Shmerkin’s theorem actually concerns a pair of δ-measures µ1, µ2 instead of δ-sets: the measures of interest for our application are simply µ1“ |A|´1H0|A and µ2“ |B|´1H0|B, and with such choices [34, Theorem 2.1] implies Theorem 4.6. Remark 4.8.We will be applying Theorem 4.6 to dyadic scales of the form δ“2´ℓmN , where ℓ, m, N PN. Since δ“ p2´mqℓN “ p2´ℓmqN, a δ-set AĂ r0,1qmay be pm, ℓNquniform, pℓm, Nq-uniform, or both. The former condition means that the branching numbers Rm Apsq “ |AXI|2´mps`1qare well-defined for mP rℓNs, whereas the latter means that the branching numbers Rℓm Apσq “ |AXI|2´ℓmpσ`1qare well-defined for σP rNs. It is clear that every pm, ℓNq-uniform 2´ℓmN -set is pℓm, Nq-uniform, and indeed Rℓm Apσq “ ℓpσ`1q´1 ź s“ℓσ Rm Apsq, σ P rNs. The converse is generally not true, so pm, ℓNq-uniformity is a strictly stronger property than pℓm, Nq-uniformity. We will also be interested in pairs pA, Bqwhich are sometimes ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 23 pη, m, ℓNq-polarised, and sometimes pη, ℓm, Nq-polarised. In contrast to uniformity, there is no simple implication between these two properties. In addition to Shmerkin’s theorem, we will also need a lemma from its proof: Lemma 4.9. Let m, ℓ, N PN,δ“2´ℓmN , and let AĂ r0,1qbe an pm, ℓNq-uniform δset. Then Ais also pℓm, N, Rℓm Aq-uniform for some RA:rNs Ñ t1,...,2ℓmu. If SĂ rNsis arbitrary, there exists A1ĂAwhich is pm, ℓNq-uniform, and also pℓm, N, Rℓm A1q-uniform with |A1| ě |A|¨ ź σPS Rℓm Apσq´1,and Rℓm A1pσq “ #1, σ PS, Rℓm Apσq, σ RS. A similar statement holds true if SĂ rℓNs, with the only difference that "Rℓm Apσq" and "Rℓm A1pσq" should be replaced by "Rm Apsq" and "Rm A1psq" for sP rℓNs. The lemma above is [34, Lemma 3.7]. To be accurate, the statement about A1remaining pm, ℓNq-uniform is not part of the statement of [34, Lemma 3.7], but the 3.8-line proof quickly reveals that pm, ℓNq-uniformity is not violated when passing between Aand A1; the only point is to "collapse" all the branching of Afor levels corresponding to σPS, or equivalently for sP tℓσ, ℓpσ`1q´1ufor all σPS. 4.3. Applying the inverse theorem. We start by fixing the following parameters: $ ’ & ’ % ℓ“ℓpα, β, γ, κq P N, η“ηpα, β, γ, κq P p0,1q, m0PNwith m0ě p40 `C0q{η. (4.10) Here C0ą0is an absolute constant to be specified later. In fact, the values of all these constants will be specified later, but as indicated above, all of them only depend on α, β, γ, κ. For the reader interested in seeing specific choices, we refer to (4.23) and the discussion afterwards. Recall that the set Bsatisfies the Frostman condition |BX Bpx, rq| ď rκ|B|for all δďrďδǫ0, where we may freely choose ǫ0“ǫ0pα, β, κ, γq ą 0. We choose ǫ0:“η. (4.11) We will assume that η, ǫ, ǫBă1{1000 in the sequel (but these upper bounds will generally not suffice). This section is devoted to the proof of the following proposition, whose proof will also finalise the choice of the parameters ǫ, ǫBą0, relative to η: Proposition 4.12. There exist ǫ, ǫBą0and měm0, depending on α, β, γ, κ, such that the following holds for all δP2´Nof the form δ“2´ℓmN ,NPN. Assume that A, B Ă r0,1s are δ-sets satisfying the small doubling assumptions (4.2). Then there exist subsets A1ĂAand B1ĂBwith the following properties: (1) A1and B1are pm, ℓNq-uniform with |A1| ě δη|A|and |B1| ě δη{2|B|. (2) The pair pA1, B1qis pη, m, ℓNq-polarised. (3) The set B1is pη{2, ℓm, Nq-polarised. Remark 4.13.In the sequel, we will always work with scales of the form δ“2´ℓmN with the fixed parameters ℓ, m, which depend on α, β, γ, κ. In other words, we initially prove Theorem 3.28 (and find the constants ǫ, ǫ0, ǫB) for only scales of this special form. After this has been accomplished, it is easy to check that the case of general scales δP2´Nis a 24 TUOMAS ORPONEN corollary, assuming that the upper bound δ0“δ0pα, β, γ, κq ą 0for δis sufficiently small. The reason is that if δP2´Nis arbitrary, then there exists a scale of the form ¯ δ“2´ℓmN with δď¯ δ.α,β,γ,κ δ. We leave the rest of this reduction to the reader. As another remark, we will later in the paper need to assume that ǫ“ǫpα, γq ą 0is sufficiently small that ǫ 1´α´ǫďγ 2.(4.14) This requirement should be combined with the one coming from Proposition 4.12. Proof of Proposition 4.12.We begin by applying Theorem 4.6 with constant η3ą0to the pair pB, Bq, for which we assumed in (4.2) that |B`B| ď δ´ǫB|B|. Assume that mě mpη3q P Nis sufficiently large that Theorem 4.6 applies. Assume additionally that mě m0, where m0is the constant from (4.10). Then, if ǫB“ǫBpη3, ℓmq “ ǫBpα, β, γ, κq ą 0 and δ“ p2´ℓmqNare sufficiently small, we find an pℓm, Nq-uniform subset B1ĂBsuch that |B1| ě δη3|B|, and B1is pη3, ℓm, Nq-polarised. We have now fixed the value of the parameter ǫBą0in (4.10)(and hence in Theorem 3.28)! Next, note that |A`B1| ď |A`B| ď δ´ǫ|A|. We therefore may apply Theorem 4.6 again to the pair pA, B1q, again with parameter η3ą0. If ǫ“ǫpη3, mq ą 0is sufficiently small, we find an pm, ℓNq-uniform subset A1ĂAwith |A1| ě δη3|A| ě δη|A|, and an pm, ℓNq-uniform subset B2ĂB1such that |B2| ě δη3|B1|,(4.15) and pA1, B2qis pη3, m, ℓNq-polarised. In particular pA1, B2qis pη, m, ℓNq-polarised. We have now fixed the value of the parameter ǫą0in (4.2)! Are we done with properties (1)-(3) in Proposition 4.12? Not quite: while passing from B1to B2, we might have lost the pη3, ℓm, Nq-polarisation of B1. The plan will be to pass to a final pm, ℓNq-uniform subset B3ĂB2which is pη{2, ℓm, Nq-polarised, and such that |B3| ě δη{4|B2|. Then finally |B3| ě δη{4|B2| ě δη{4`η3|B1| ě δη{4`2η3|B| ě δη{2|B|. Also pA1, B3qremains pη, m, ℓNq-polarised, since this property is not violated by replacing B2by an pm, ℓNq-uniform subset, for example B3. Write S0:“ tσP rNs:Rℓm B1pσq “ 1uand S1:“ tσP rNs:Rℓm B1pσq ě 2p1´η3qℓmu. Since B1was constructed to be pη3, ℓm, Nq-polarised, we have rNs “ S0YS1, and |B1| “ ź σPS1 Rℓm B1pσq ě 2p1´η3qℓm|S1|. Now, let Sbad :“ tσPS1:Rℓm B2pσq ă 2p1´η{2qℓmu, and Sgood :“S1zSbad. (Note that the numbers Rℓm B2pσqare well-defined, since B2is pm, ℓNq-uniform, hence pℓm, Nq-uniform.) Then, since evidently Rℓm B2pσq ď Rℓm B1pσq “ 1for all σPS0, we have |B2| “ ź σPSbad Rℓm B2pσq¨ ź σPSgood Rℓm B2pσq ď 2p1´η{2qℓm|Sbad |¨2ℓmp|S1|´|Sbad|q “2ℓm|S1|´pη{2qℓm|Sbad|. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 31 where Γwas defined in (4.29). In particular, ξ&α,β,γ 1. We also impose the following additional condition on the constant ℓPNselected at (4.10): ℓěξ´1“4 γ´Γ.(4.37) Recall that J“ tt´r,...,tu P N`was the shortest extension (to the left) of a certain interval J0PℓNB1with the property Rm B1pJq ě 2ζm|J|“2ζmpr`1q. Consequently, the subinterval Jξ“ tt´tp1´2ξqru,...,tudoes not yet have this property, that is, Rm B1pJξq ă 2ζm|Jξ|“2ζmptp1´2ξqru`1q.(4.38) Here we used that tp1´2ξqruăr, which is true because ξr ěξℓ ě1by (4.37). Now, it follows from a combination of (4.38), and Rm B1pJq ě 2ζm|J|“2ζmpr`1q, that Rm B1pJzJξq ě 2ζmpr´tp1´2ξqruqě22ξζmr ě2ξζmpr`1q“2ξζm|J|.(4.39) We are then prepared to define the desired subset B2ĂB1. Let Sξ:“ YtJξ:JPN`u, and apply the "collapsing" Lemma 4.9 to the pm, ℓNq-uniform set B1, and the set of scales SξĂ rℓNs. The product is an pm, ℓNq-uniform subset B2ĂB1such that Rm B2psq “ #Rm B1psq, s RSξ, 1, s PSξ. In particular, Rm B2psq “ Rm B1psqfor all sPJzJξ, for JPN`, so (4.39) remains valid for the set B2: Rm B2pJq ě Rm B1pJzJξq ě 2ξζm|J|.(4.40) In fact, the first inequality is an equation, since Rm B2psq “ 1for all sPJξĂSξ. Curiously, we will have no use for a "global" lower bound for |B2|, although it would be easy to deduce from (4.28) that |B2| ě δ2ζ¨ |B1|. From now on, only the "local" branching estimate (4.40) will be needed, and "global" lower bound |B1|'δ´βhas already been fully exploited in previous sections (where the relation between γ, α and βappeared). The point of reducing B1to B2was to improve the 2´mpt`1q-separation of distinct intervals IPImpt`1qpB1qto something resembling 2´mpt´rq-separation. This has now been accomplished. More precisely, assume that J“ tt´r,...,tu P N`, let IPImpt´rqpB2q, and and let I1, I2PImpt`1qpB2qbe distinct. Then, since Rm B2psq “ 1, s PJξ“ tt´tp1´ξqru,...,tu, the intervals I1, I2are contained inside distinct intervals ˆ I1,ˆ I2PImpt´tp1´ξqruqpB2q Ă Impt´tp1´ξqruqpB1q. Consequently, using also that tp1´ξqruě p1´ξqr´1, and ξr ěξℓ ě1, distpI1, I2q ě distpˆ I1,ˆ I2q(4.17) ě2´mpt´tp1´ξqruq ě2´ξmr´m¨2´mpt´rq ě2´2ξmpr`1q¨2´mpt´rq.(4.41) Inequality (4.41) is more clearly phrased in the following way: 32 TUOMAS ORPONEN Lemma 4.42. Let J“ tt´r,...,tu P N`,∆J:“2´mpt´rq, and δJ:“2´mpt`1q. Let IPImpt´rqpB2qbe a dyadic interval of length ∆Jintersecting B2, and let I1, I2PImpt`1qpB2q be distinct with I1, I2ĂI. Then, distpI1, I2q ě ˆδJ ∆J˙2ξ ¨|∆J|. Proof. Observing that δJ{∆J“2´mpr`1q, this inequality is just a rewording of (4.41).  4.8. Elementary projection estimates. The plan is to prove lower bounds for |A1`cB2|δ by, roughly speaking, establishing separately lower bounds for |pA1XIq` cpB2XJq|δ, where I, J Ă r0,1qare suitable dyadic intervals intersecting A1, B2, and then combining the results. In this section, we will prove an auxiliary result which will imply the required lower bounds for |pA1XIq`cpB2XJq|δ. To be more accurate, instead of proving lower bounds for |pA1XIq ` cpB2XJq|δdirectly, we prove (stronger) lower bounds for the entropies of suitable measures supported on pA1XIq ` cpB2XJq(see (4.58)). This is (only!) done for the reason that such "multi-scale" information about entropy is cleaner to combine than "multi-scale" information about cardinalities. We introduce the following notation. Dyadic cubes in Rdof side-length 2´nare denoted Dn. If µis a Borel probability measure on Rd, and nPN, we write µpnq:“ÿ QPDn µpQq LdpQq¨Ld|Q. Thus µpnqis a "2´n-discretisation of µ". Note that µpnqPL2pRdqXL8pRdq. We also define the projections πcpx, yq:“x`cy for px, yq P R2and cPR. Lemma 4.43. Let ∆“2´nP2´N, and let γ, γA, γBP p0,1s, and Cě1. Let A,BĂDnbe collections of dyadic ∆-intervals with |A| “ ∆´γAand |B| “ ∆´γB. We assume the following separation from B, for some ξP p0,1s: distpI1, I2q ě ∆ξfor distinct I1, I2PB.(4.44) Let ‚Let µbe a probability measure with spt µĂ pYAqˆpYBqwith the property that µpQq ď C∆γA`γBfor QPDn. ‚Let νbe a probability measure on r´1,1ssuch that νpIq ď C∆γfor all IPDn. Then, ż1 ´1}pπcµqpnq}2 2dνpcq.C¨maxt∆γA`γB´1,∆γ´1´ξu.(4.45) Remark 4.46.To help interpreting the upper bound (4.45), let us mention the "trivial" estimate }pπcµqpnq}2 2.∆γA´1for every cP r0,1q. This could be deduced rather easily from (4.47) below. Therefore, (4.45) beats the trivial bound whenever γąγA`ξ. Proof of Lemma 4.43.By definition, pπcµqpnq“ÿ IPDn πcµpIq ∆¨L1|I“ÿ IPDn µpπ´1 cpIqq ∆¨L1|I. c P r´1,1s. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 33 Consequently, }pπcµqpnq}2 2“ÿ IPDnˆµpπ´1 cpIqq ∆˙2 ¨∆“1 ∆¨ÿ IPDnpµˆµqptpp, qq:p, q Pπ´1 cpIquq.(4.47) Therefore, ∆¨ż1 ´1}pπcµqpnq}2 2dνpcq „ ż1 ´1ÿ IPDnpµˆµqptpp, qq:p, q Pπ´1 cpIquqdνpcq “¨żÿ IPDn 1tp,qPπ´1 cpIqupcqdνpcqdµppqdµpqq. We split the outer integration into Ωnear :“ tpp, qq:|p´q| ă 10∆uand Ωfar :“ tpp, qq:|p´q| ě 10∆u. Regarding Ωnear, we only use the observation that if p, q PAˆBand cP r0,1sare fixed, then there is at most one interval IPDnsuch that p, q Pπ´1 cpIq. Since µ, ν are probability measures, and µpBpx, 10∆qq .C∆γA`γBfor every xPR2, this leads to ¨Ωnear ÿ IPDn 1tp,qPπ´1 cpIqupcqdνpcqdµppqdµpqq.pµˆµqpΩnearq.C∆γA`γB.(4.48) We then consider integral over the domain Ωfar. A basic, easy to verify, observation is this: if p, q PR2are fixed and distinct, then the set Ipp, qq:“ tcP r0,1s:p, q Pπ´1 cpIqfor some IPDnu is contained in an interval of length .∆{|p´q|, and in particular can be covered by .|p´q|´1dyadic intervals of length ∆. We combine this with the following additional observation. Note that all the tubes π´1 cpIqmake an angle ďπ{4with the y-axis (this is attained for c“1, and for c“0, the tubes π´1 cpIqare vertical). Therefore, p, q PAˆB, |p´q| ě 10∆ and DcP r´1,1ss.t. p, q Pπ´1 cpIq ùñ |py´qy| ą ∆. Here py, qyPBrefer to the second coordinates of p, q. Namely, if |p´q| ě 10∆ and |py´qy| ă ∆, then |px´qx| ě 9∆, which makes the pair p, q too "horizontal" to be contained in any common tube π´1 cpIq, with cP r´1,1sand IPDn. Now, recalling our assumption (4.44) that distpI1, I2q ě ∆ξfor distinct I1, I2PB, the conclusion |py´qy| ą ∆ can be amplified substantially: |py´qy| ą ∆implies that py, qylie in distinct intervals in B, hence |p´q| ě |py´qy| ě ∆ξ. Therefore: ¨Ωfar żÿ IPDn 1tp,qPπ´1 cpIqupcqdνpcqdµppqdµpqq “ ¨ΩFAR żIpp,qq . . . dνpcqdµppqdµpqq, with ΩFAR “ tpp, qq:|p´q| ě ∆ξu. Now, for every pair pp, qq P ΩFAR, we note that the set Ipp, qq Ă r0,1scan be covered by .|p´q|´1ď∆´ξdyadic intervals of length ∆, and for each cPIpp, qq, there is exactly one IPDnsuch that p, q Pπ´1 cpIq. Therefore, żIpp,qqÿ IPDn 1tp,qPπ´1 cpIqupcqdνpvq “ νpIpp, qqq .C∆γ´ξ,pp, qq P ΩFAR, 34 TUOMAS ORPONEN and consequently ¨Ωfar żÿ IPDn 1tp,qPπ´1 cpIqupcqdνpcqdµppqdµpqq.C∆γ´ξ. Combining this estimate with (4.48), we arrive at (4.45).  We will next deduce, as a corollary, an entropy version of Lemma 4.43. For this purpose, we record the following [35, Lemma 3.6] by Shmerkin: Lemma 4.49. Let µbe a Borel probability measure on Rd. The following relation holds between the Dn-entropy Hpµ, Dnqof µ, and the L2-norm of µpnq: Hpµ, Dnq ě dn ´log }µpnq}2 2.(4.50) Here, and below, "log" refers to logarithm in base 2. Corollary 4.51. Let ∆“2´nP2´N, and assume that A,B, µ, ν, γA, γB, γ, C, and ξhave the same meaning as in Lemma 4.43. Then, ż1 ´1 Hpπcµ, Dnqdνpcq ě n¨mintγA`γB, γ ´ξu´log C´log C0,(4.52) where C0ą0is an absolute constant. Proof. First combine (4.50) (with d“1) and Jensen’s inequality to deduce that ż1 ´1 Hpπcµ, Dnqdνpcq ě n´ż1 ´1 log }pπcµqpnq}2 2dνpcq ě n´log ˆż1 0}pπcµqpnq}2 2dνpcq˙. Here, ż1 ´1}pπcµqpnq}2 2dνpcq ď C0Cmaxt2np1´γA´γBq,2np1`ξ´γqu for some absolute constant C0ą0, by Lemma 4.43. These inequalities give (4.52).  4.9. Projecting pieces of A1ˆB2.We next put Corollary 4.51 to work in our "real-world" situation. We recall the following notation from Section 2.1. Assume that µis a Borel probability measure on Rd(we will use this for both d“1and d“2), and let QPDnbe a dyadic cube of side-length 2´nsuch that µpQq ą 0. Let TQ:QÑ r0,1qdbe the rescaling map with TQpQq “ r0,1qd. We define the measures µQ:“1 µpQq¨µ|Qand µQ:“TQµQ.(4.53) In this section, µ“µA1ˆµB2, where µA1is the normalised counting measure on A1, and µB2is the normalised counting measure on B2(defined in Section 4.7). For sP rℓNs, we will write Dmspµq:“ tIˆJ:IPImspA1qand JPImspB2qu “ tQPDms :µpQq ą 0u. Fix J“ tt´r,...,tu P Nlow `. As defined in (4.30), this means that Rm A1pJq ď 2Γm|J|, where ΓP ppα´βq{p1´βq, γq Ă rγ{2, γqwas the parameter specified in (4.29). For now, it is only important to remember that γ´Γ&α,β,γ 1. Fix intervals I0PImpt´rqpA1qand J0PImpt´rqpB2q. Write AI0:“ tI1PImpt`1qpA1q:I1ĂI0uand BJ0:“ tJ1PImpt`1qpB2q:J1ĂJ0u. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 35 Then |AI0| “ Rm A1pJq ď 2Γm|J|and |BJ0| “ Rm B2pJq(4.40) ě2ξζm|J|.(4.54) In particular, we may write Rm A1pJq “ |AI0| “ 2γAm|J|and |BJ0| “ 2γBm|J|(4.55) for some 0ďγAďΓand γBěξζ. Then, write Q0:“IˆJPDmpt´rqpµq, n :“m|J| “ mpr`1qand ∆ :“2´n“2´mpt`1q 2´mpt´rqěδ. Consider the normalised measure µQ0, as in (4.53). The measure µQ0is supported on a product of the form pYAqˆpYBq “ TQ0ppYAI0qˆpYBJ0qq, where A,Bare the families of ∆-intervals obtained by normalising the intervals in AI0and BJ0by a factor of 2mpt´rq. It follows from the pm, ℓNq-uniformity of A1and B2that µQ0pQq ď p|A||B|q´1“ p|AI0||BJ0|q´1(4.55) “∆γA`γB, Q PDn. Moreover, the intervals in Bsatisfy the following separation property by Lemma 4.42: I1, I2PB, I1‰I2ùñ distpI1, I2q ě ∆2ξ. These facts place us in a position to apply Corollary 4.51 to the measure µQ0: ż1 ´1 HpπcµQ0,Dnqdνpcq ě n¨mintγA`γB, γ ´2ξu´log 40 ´log C0.(4.56) The parameter "ξ" was initially chosen (see (4.36)) so that 2ξď pγ´Γq{2. Since γAďΓ, this leads to γ´2ξěγA`pγ´Γq´2ξěγA`pγ´Γq{2. Recalling also that γBěξζ by (4.54), and ζP p0,1q(see (4.22) for a reminder), we find mintγA`γB, γ ´2ξu ě mintγA`ξζ, γA`pγ´Γq{2u “ γA`ξζ. (4.57) Before the final conclusion, let us recall that n“mpr`1q “ m|J|, and observe that γA¨n“γA¨m|J|(4.55) “log Rm A1pJq. Therefore, (4.56)-(4.57) yield ż1 ´1 HpπcµQ0,Dm|J|qdνpcq ě n¨pγA`ξζq´log 40 ´log C0 “log Rm A1pJq`ξζ ¨m|J|´log 40 ´log C0(4.58) for all J“ tt´r,...,tu P Nlow `and for all Q0“IˆJPDmpt´rqpµq. 36 TUOMAS ORPONEN 4.10. Final multiscale argument. As in the previous section, let µA1be the normalised counting measure on the set A1, let µB2be the normalised counting measure on the set B2, and let µ“µA1ˆµB2. Recall also that Dmspµq “ tQPDms :µpQq ą 0ufor sP rℓNs. We warn the reader that the notation "Dms" will in this section refer to both dyadic squares in R2, and dyadic intervals in R. The meaning should always be clear from context. The purpose fo this section is to show that there exists cPsptpνqsuch that 1 ℓmN ¨Hpπcµ, DℓmN q ě ¯α`ξζ ¨1 2rp1´βq´pα´βq{Γs´2η. (4.59) Here ¯αwas the constant (defined in (4.1)) such that |A| “ δ´¯α. The lower bound in (4.59) yields a lower bound for |A1`cB2|δ, and consequently |A`cB|δ: since Hpπcµ, DℓmN q ď log |A1`cB2|δďlog |A`cB|δ, and ℓmN “ ´log δ, we deduce from (4.59) that log |A`cB|δ ´log δě¯α`ξζ ¨1 2rp1´βq´pα´βq{Γs´2η. If ηą0is sufficiently small, depending only on α, β, γ, κ, this implies |A`cB|δě δ´¯α´η“δ´η|A|. Of course it is important here that the values of ξ“ pγ´Γq{4(see (4.36)) and ζą0(see (4.22)) are independent of ¯α, although they may depend on α. This proves Theorem 3.28: either (4.2) fails, and |A`cB|δěδ´ǫ|A|with c“1Psptpνq, or (4.2) holds, and in this case |A`cB|δěδ´η|A|for the point cPsptpνqprovided by (4.59). It remains to prove (4.59). This will be accomplished by combining (4.58) with the following uniform lower bound: Lemma 4.60. Let J“ ts, . . . , tu Ă rℓNs, and let Q0PDmspµq. Then, HpπcµQ0,Dm|J|q ě log Rm A1pJq´1, c P r0,1s.(4.61) Proof. Let IPImspA1qand JPImspB2qsuch that Q0“IˆJ. Then µQ0“µI A1ˆµJ B2, hence πcµQ“µI A1˚µJ B2, and finally HpπcµQ0,Dm|J|q “ HpµI A1˚µJ B2,Dm|J|q ě żHppµI A1qx,Dm|J|qdµJ B2pxq,(4.62) where the inequality follows from the concavity of entropy (we discussed this at (2.5)), and where pµA1qI xpHq:“µI A1pH´xqfor HĂR. From the definition of entropy, one has HppµI A1qx,Dm|J|q “ HpµI A1,Dm|J|´xq, where Dm|J|´xrefers to the family of p´xq-translated dyadic intervals. Now, for xPR fixed, every intervals in Dm|J|can be covered by 2intervals in Dm|J|´xand vice versa. This implies that |HpµI A1,Dm|J|´xq´HpµI A1,Dm|J|q| ď log 2 “1, x PR.(4.63) Furthermore, by definition, HpµI A1,Dm|J|q “ ´ ÿ LPDm|J| µI A1pLqlog µI A1pLq. Since A1is pm, ℓN, Rm A1q-uniform, either µI A1pLq “ 0, or then µI A1pLq “ Rm A1pJq´1for every LPDm|J|. Therefore HpµI A1,Dm|Jq “ Rm A1pJq´1. In combination with (4.62)-(4.63), this yields (4.61).  ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 37 Recall the intervals Nlow `ĂN`, defined in (4.30). In this section, the properties of these intervals will be used via the formula (4.58), and we additionally need to recall that ÿ JPNlow ` |J| ě 1 2rp1´βq´pα´βq{Γs¨ℓN (4.64) by (4.31). Let Pbe the partition of rℓNswhich is induced by the intervals in Nlow `. In other words, Pconsists of the intervals in Nlow `, and the maximal complementary intervals. We write Puseless :“PzNlow `, and we enumerate P“ tJ1,J2,...,Jhu, where 1ďhďℓN. We write Jj“ tsj,...,tju for 1ďjďh, so s1“0,th`1“ℓN, and sj`1“tj`1for all 1ďjăh. We artificially define sh`1:“ℓN, so the relation sj`1“tj`1also remains valid for j“h. We abbreviate Dj:“Dmsjpµq:“ tIˆJ:IPImsjpA1qand JPImsjpB2qu. We then apply the entropy lower bound in Lemma 2.3, and its corollary (2.4), to the partition 0“ms1ă... ămshămsh`1“ℓmN of t0, . . . , ℓmNu, and the 2-Lipschitz maps πc:R2ÑRwith cP r´1,1s: ż1 ´1 Hpπcµ, DℓmN qdνpcq “ h ÿ j“1ÿ QPDj µpQqż1 ´1 HpπcµQ,Dmsj`1´msj|D0qdνpcq ě ´C0h` h ÿ j“1ÿ QPDj µpQqż1 ´1 HpπcµQ,Dmptj´sj`1qqdνpcq.(4.65) Above, tj´sj`1“ |Jj|. For JjPNlow `, and QPDj, we recall from (4.58) that ż1 ´1 HpπcµQ,Dm|Jj|qdνpcq ě log Rm A1pJjq`ξζ ¨m|Jj|´log 40 ´log C0. For JPPuseless we have to settle with the estimate ż1 0 HpπcµQ,Dm|Jj|qdνpcq ě log Rm A1pJjq´1 from Lemma 4.60. Plugging these bounds into (4.65) (and redefining C0as C0`1) yields ż1 ´1 Hpπcµ, DℓmN qdνpcq ě ´p40 `C0qh`ÿ JPP log Rm A1pJq`ξζ ÿ JPNlow ` |J| (4.64) ělog |A1|`ξζ ¨1 2rp1´αq´pα´βq{Γs¨ℓmN ´hp40 `C0q. Recalling that |A1| ě δη|A| ě 2p¯α´ηqℓmN , there exists cPsptpνqwith HℓmN pπcµq ě `¯α`ξζ ¨1 2rp1´αq´pα´βq{Γs´η˘´hp40`C0q ℓmN Here hp40 `C0q{pℓmNq ď p40 `C0q{m0ďηby the choice of m0at (4.10), and since we chose měm0in Proposition 4.12. Therefore we have established (4.59), and completed the proof of Theorem 3.28. 38 TUOMAS ORPONEN 5. HAUSDORFF DIMENSION ESTIMATES The purpose of this final section is to reduce Theorem 1.6 to Theorem 1.8, and to use Theorem 1.6 to prove the Hausdorff dimension result, Corollary 1.7. Remark 5.1.The threshold γą pα´βq{p1´βqfamiliar from Theorems 1.6 and 1.8 plays no particular role in this section: if we knew that Theorem 1.8 holds for all γP pτ, 1sfor some parameter τ“τpα, βq P p0,1q, then the argument would below would show that Theorem 1.6 also holds for γąτ. This is relevant to know if one eventually manages to solve Conjecture 1.5, and proves Theorem 1.8 with threshold τpα, βq “ α´β. 5.1. Reducing Theorem 1.6 to Theorem 1.8: outline. The reduction from Theorem 1.6 to Theorem 1.8 proceeds in several stages. First, in Section 5.2, we prove the following toy version of Theorem 1.6: instead of allowing for general subsets of the form GĂ AˆBwith |G| ě δǫ|A||B|, this version (Theorem 5.3) only treats subsets of the form G“AˆB1with |B1| ě δǫ|B|. The conclusion is that there exists cPsptpνqsuch that |A`cB1| ě δ´ǫ|A|for all B1ĂBwith |B1| ě δǫ|B|. Even the toy version, Theorem 5.3, is not proved directly: we will pass through a toytoy version, Theorem 5.4, where we are first allowed to replace AˆBby a subset of the form Aˆ¯ B, and then the conclusion explained above is established for Aˆ¯ Bin place of AˆB. Fortunately, the passage between the toy and toy-toy versions can be accomplished by a formal exhaustion argument, which I learned from He’s paper [17]. The toy-toy version is eventually deduced, in Section 5.4, by a direct argument from the main Theorem 1.8. This is the heart of the matter. Instead of giving details here, I mention a key difficulty: this reduction, and various other steps of the argument would be simpler if we a priori knew that |A`A| « |A|and |B`B| « |B|.(5.2) (In this heuristic discussion, I will leave the meaning of "«" to the reader’s imagination.) In the case |A| « |B|, treated by Bourgain in [5], this is automatic: if |A`cB|δ« |A| « |B| for some cP r1 2,1s, then (5.2) holds by Plünnecke’s inequality. However, in our situation Bis typically much smaller than A, and now the property |A`cB|δ« |A|implies neither property in (5.2). Nevertheless, (5.2) is needed, technically because Lemma 5.16 is useless without (5.2). Roughly speaking, Theorem 5.4 is proved by making a counter assumption, and using it to generate new sets ¯ A‰Aand ¯ B‰Bwhich satisfy the original hypotheses, and additionally (5.2). At some level, this argument is reminiscent of the proof of the asymmetric Balog-Szemerédi-Gowers theorem in [40] (see Theorem 5.38). Once we have the toy version, Theorem 5.3, at our disposal, it remains to deduce Theorem 1.6 from Theorem 5.3. This step is based on the asymmetric Balog-SzemerédiGowers theorem – unlike the other steps. We make a counter assumption that for every cPsptpνqthere exists a subset GcĂAˆBwith |G|'|A||B|such that |πcpGq|δ/|A|. By the B-S-G theorem, this yields for every cPsptpνqsubsets AcĂAand BcĂBsuch that |Ac|'|A|,|Bc|'|B|, and |Ac`cBc|δ/|A|. With the help of probabilistic arguments, and the Plünnecke-Ruzsa inequality (Lemma 3.3), this allows us to construct a new δseparated set HĂ r0,1swith |H|/|A|, and a subset CĂsptpνqwith νpCq'1, such that |H`cBc|δ/|H|for all cPC. This violates the first toy version, Theorem 5.3, applied to H, B and finally concludes the proof of Theorem 1.6. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 39 5.2. A toy version. Theorem 1.6 claims the existence of cPsptpνqsuch that |πcpGq| ě δ´ǫ|A|for all GĂAˆBwith |G| ě δǫ|A||B|. A toy problem is to find cPsptpνqsuch that |A`cB1|δěδ´ǫ|A|for all B1ĂBwith |B1| ě δǫ|B|. Instead of approaching Theorem 1.6 directly, we will first solve this toy problem: Theorem 5.3. Let 0ăβďαă1and κą0. Then, for every γP ppα´βq{p1´βq,1s, there exist ǫ0, ǫ, δ0P p0,1 2s, depending only on α, β, γ, κ, such that the following holds. Let δP2´N with δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1ssatisfy the following hypotheses: (A) |A| ď δ´α. (B) |B| ě δ´β, and Bsatisfies the following Frostman condition: |BXBpx, rq| ď rκ|B|, δ ďrďδǫ0. Further, let νbe a Borel probability measure with sptpνq Ă r0,1s, and satisfying the Frostman condition νpBpx, rqq ď rγfor xPRand 0ărďδǫ0. Then, there exists cPsptpνqsuch that if B1ĂBsatisfies |B1| ě δǫ|B|, then |A`cB1| ě δ´ǫ|A|. 5.3. Reduction to a weaker toy theorem. Even Theorem 5.3 is hard to prove with a direct assault. We will first need to reduce it to an even weaker version. In the statement, we use the following notation (slightly adapted) from He’s paper [17]. Given two sets A, B Ă r0,1sXpδ¨Zq, we write EpA|B, ǫq:“ tcPR:DB1ĂBsuch that |B1| ě δǫ|B|and |A`cB1|δăδ´ǫ|A|u. Theorem 5.4. Let 0ăβďαă1and κ, θ ą0. Then, for every γP ppα´βq{p1´βq,1s, there exist ǫ0, ǫ, δ0P p0,1 2s, depending only on α, β, γ, κ, such that the following holds. Let δP2´N with δP p0, δ0s, and let A, B Ă pδ¨ZqXr0,1ssatisfy the following hypotheses: (A) |A| ď δ´α. (B) |B| ě δ´β, and Bsatisfies the following Frostman condition: |BXBpx, rq| ď rκ|B|, δ ďrďδǫ0. Further, let νbe a Borel probability measure with sptpνq Ă r0,1s, and satisfying the Frostman condition νpBpx, rqq ď rγfor xPRand 0ărďδǫ0. Then, there exists a subset B1ĂBsuch that νpEpA|B1, ǫqq ď δǫ. I learned this reduction from the paper of He [17, Proposition 25], and his proof works here, up to modifying the notation. The full details are recorded below nonetheless. Proof of Theorem 5.3 assuming Theorem 5.4.Let α, β, γ, κ be the parameters given in Theorem 5.3, so that γą pα´βq{p1´βq. Our task is to find the constants ǫ, ǫ0, δ0P p0,1 2s, depending only on α, β, γ, κ. Start by applying Theorem 5.4 with parameters α, ¯ β, γ, ¯κ, where ¯κP p0, κqis arbitrary, and also and ¯ βăβis arbitrary with the property that the key inequality γą pα´¯ βq{p1´¯ βq remains valid. Let ¯ǫ, ¯ǫ0,¯ δ0P p0,1 2sbe the constants given by Theorem 5.4, associated to the parameters α, ¯ β, γ, ¯κ. We define ǫ0:“¯ǫ0and ǫ:“min "¯ǫ 2,pκ´¯κq¯ǫ0 4,β´¯ β 2*.(5.5) 40 TUOMAS ORPONEN We assume that δ0ď¯ δ0, and there will be a few additional requirements, where for example δďδ0needs to be taken small enough relative to the difference ¯ǫ´ǫ. I will not gather these requirements together; they will be pointed out where they appear. Let δP2´Nwith δďδ0, and let A, B, ν be the objects from Theorem 5.3, satisfying the assumptions of that theorem with constants α, β, κ, γ, and ǫ0, δ0as above. In particular, |B| ě δ´βand |BXBpx, rq| ď rκ|B|for xPRand δďrďδǫ0.(5.6) Evidently A, B, ν also satisfy the hypotheses of Theorem 5.4 with constants α, ¯ β, γ, κ{2, and ¯ǫ0. We now perform an "exhaustion" argument to construct a finite sequence of disjoint subsets B1,...,BNĂBwith the property νpEpA|Bj,¯ǫqq ď δ¯ǫ,1ďjďN. (5.7) Let B1ĂBbe the set given initially by Theorem 5.4. We then assume inductively that we have already constructed disjoint B1,...,BnĂBfor some ně1. There are two options: ˇˇˇBz n ď j“1 Bjˇˇˇăδ2ǫ|B|or ˇˇˇBz n ď j“1 Bjˇˇˇěδ2ǫ|B|.(5.8) In the former case, the inductive construction terminates, and we define N:“n. In the latter case, we apply Theorem 5.4 to the objects A, ν, and B1:“BzŤn j“1Bj. This is legitimate, because |B1| ě δ2ǫ|B| ě δ´β´2ǫěδ´¯ β, and |B1XBpx, rq| (5.6) ďrκ|B| ď δ´2ǫrκ|B1|(5.5) ďr¯κ|B1|, x PR, δ ďrďδǫ0“δ¯ǫ0. Therefore A, B1, ν satisfy the hypotheses of Theorem 5.4 with constants α, ¯ β, ¯κ, γ, ¯ǫ0. Consequently, there exists a further subset Bn`1ĂB1“BzŤn j“1Bjwith the property νpEpA|Bn`1,¯ǫqq ď δ¯ǫ. This completes the inductive construction of the sequence B1,...,BN. The construction terminates in ďδ´¯ǫsteps, because the sets Bjsatisfy |Bj| ě δ´¯ǫ. Indeed, since νpEpA|Bj,¯ǫqq ă 1, there exists cPsptpνqzEpA|Bj,¯ǫq, and then |A||Bj| ě |A`cBj|δěδ´¯ǫ|A|. When the inductive procedure eventually terminates, we write B0:“ŤN j“1Bj. By (5.8), we have |BzB0| ă δ2ǫ|B|. Now, note that the claim of Theorem 5.3 is equivalent to proving that sptpνqzEpA|B, ǫq ‰ H. We will prove this by showing that EpA|B, ǫqhas small νmeasure. The first step is to establish the following inclusion: EpA|B, ǫq Ă ď Jč jPJ EpA|Bj,¯ǫq,(5.9) where the index set Jruns over all subsets of t1,...,Nuwith řjPJ|Bj| ě δǫ|B|{4. The proof is nearly verbatim the same as in [17, Proposition 25], but I record the details here for completeness. If cPEpA|B, ǫq, then by definition there exists a subset BcĂBwith |Bc| ě δǫ|B|and |A`cBc|δăδ´ǫ|A|. Let J:“ t1ďjďN:|BcXBj| ě δ¯ǫ|Bj|u. Then cPEpA|Bj,¯ǫqfor all jPJ, since B1 j:“BcXBjĂBjsatisfies |B1 j| ě δ¯ǫ|Bj|and |A`cB1 j|δăδ´ǫ|A| ď δ´¯ǫ|A|. This proves (5.9), once we verify that řjPJ|Bj| ě δǫ|B|{4. To see this, recall that |BzB0| ď δ2ǫ|B|. This implies that Bchas large intersection with B0(assuming that δą0is sufficiently small): |BcXB0| ě 1 2¨δǫ|B|. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 47 Remark 5.37.In order to deduce Theorem 5.4 for a fixed exponent "κ" from Theorem 5.17, the argument above only needed to apply Theorem 5.17 with exponent ¯κăκarbitrarily close to κ(but ǫÑ0in Theorem 5.4 as ¯κÕκ). 5.5. Proof of the main theorem. In this section, we finally prove Theorem 1.6 by reducing it to its toy version, Theorem 5.3. We will need the asymmetric Balog-SzemerédiGowers theorem, see the book of Tao and Vu, [40, Theorem 2.35]. We state the result in the following slightly weaker form (following Shmerkin’s paper [34, Theorem 3.2]): Theorem 5.38 (Asymmetric Balog-Szemerédi-Gowers theorem).Given ζą0, there exists ξą0such that the following holds for δP2´Nsmall enough. Let A, B Ă pδ¨ZqXr0,1sbe finite sets, and assume that there exist cP r1 2,1sand GĂAˆBsatisfying |G| ě δξ|A||B|and |tx`cy :px, yq P Gu|δ“ |πcpGq|δďδ´ξ|A|.(5.39) Then there exist subsets A1ĂAand B1ĂBwith the properties |A1||B1| ě δζ|A||B|and |A1`cB1|δďδ´ζ|A|.(5.40) Remark 5.41.In the references for Theorem 5.38 cited above, the assumption |πcpGq|δď δ´ξ|A|in (5.39) is replaced by |π1pGq| ď δ´ξ|A|, and the conclusion (5.40) is replaced by |A1`B1| ď δ´ζ|A|. For cP r1 2,1s, it is easy to see that the two variants of the theorem are formally equivalent. The details are left to the reader. The idea is to begin by applying the standard version of Theorem 5.38 to the sets Bc:“ pcBqδĂδ¨Zand Gc:“ tpx, pcyqδq: px, yq P Gu Ă AˆBc, which satisfy |Bc| „ |B|,|Gc| „ |G|, and |π1pGcq| .δ´ξ|A|. Proof of Theorem 1.6 assuming Theorem 5.3.Let α, β, γ, κ be the constants for which we are supposed to prove Theorem 1.6. Thus γą pα´βq{p1´βq. Our task is to find the constants ǫ, ǫ0, δ0P p0,1 2ssuch that the conclusion of Theorem 1.6 holds. To this end, pick ¯κP p0, κqarbitrarily, and ¯αąα,¯ βăβ, and ¯γăγin such a way that the key inequality ¯γą p¯α´¯ βq{p1´¯ βq persists. This can be done explicitly in such a way that ¯α, ¯ β, ¯γare functions of α, β, γ: therefore, any future dependence on ¯α, ¯ β, ¯γwill, in fact, be a dependence on α, β, γ. Let ¯ǫ, ¯ǫ0,¯ δ0P p0,1 2sbe the constants given by Theorem 5.3 applied with parameters ¯α, ¯ β, ¯γ, ¯κ. We now define ǫ, ǫ0, δ0based on ¯ǫ, ¯ǫ0,¯ δ0. First, we set ǫ0:“¯ǫ0. We also fix δ0P p0,¯ δ0s. There will be a few additional requirements on δ0, depending on α, β, γ, κ only. These will be clarified when they arise. We then finally determine the constant ǫ. First, we fix a natural number N„1{¯ǫ, sufficiently large that the following holds: pN´1q´1ă¯ǫ{2.(5.42) Then, we fix the auxiliary constant ζ:“min "¯ǫ 20N,¯ǫ0pκ´¯κq 2N,¯α´α 2NpN`1q,β´¯ β 2N,ǫ0pγ´¯γq 2N*.(5.43) Now, let ǫ:“ξpζq ą 0be the constant given by Theorem 5.38 applied with the constant ζą0from (5.43). This means that if cP r1 2,1s, and GĂAˆBsatisfies |G| ě δǫ|A||B| and |πcpGq|δďδ´ǫ|A|, then there exist A1ĂAand B1ĂBas in (5.40). 48 TUOMAS ORPONEN Armed with these choices of parameters, we are prepared to prove Theorem 1.6. Fix δP2´Nwith δďδ0, and let A, B, ν be a triple satisfying the hypotheses of Theorem 1.6 with constants α, β, γ, κ. To recap once more, |A| ď δα, and |B| ě δ´β, and |BXBpx, rq| ď rκ|B|, x PR, δ ďrďδǫ0“δ¯ǫ0.(5.44) Also, νis a probability measure on r1 2,1ssatisfying νpBpx, rqq ď rγfor all δďrďδǫ0. We claim that there exists cPC:“sptpνqsuch that whenever GĂAˆBis a subset with |G| ě δǫ|A||B|, then |πcpGq|δěδ´ǫ|A|. We make a counter assumption: the property above fails for every cPC. Then, by the choice ǫ“ξpζq, and Theorem 5.38, for every cPCthere exist subsets AcĂAand BcĂB, for every cPC, with the properties |AcˆBc| ě δζ|A||B|and |Ac`cBc|δďδ´ζ|A|.(5.45) We observe that ż...ż|pAc1ˆBc1qX...XpAcNˆBcNq|dνpc1q¨¨¨dνpcNq ě δNζ |A||B| by Hölder’s inequality. Using pAˆBqXpCˆDq “ pAXCqˆpBXDq, and Chebyshev’s inequality, and νpRq “ 1, it follows that the set Ω :“ tpc1,...,cNq P CN:|pAc1X...XAcNqˆpBc1X...XBcNq| ě 1 2δNζ |A||B|u (5.46) satisfies νNpΩq ě 1 2¨δNζ (5.47) For c1,...,cnPCfixed, we define Ωc1¨¨¨cn:“ tpcn`1,...,cNq P CN´n:pc1,...,cNq P Ωu. It follows easily from Fubini’s theorem that νN´npΩc1¨¨¨cnq “ żνN´n´1pΩc1¨¨¨cncqdνpcq(5.48) for all c1,...,cnPC, and 1ďnďN´2. The same remains true for n“0, if the left hand side is interpreted as νNpΩq, and c1¨¨¨cnc“c. Equation (5.48) also remains valid for n“N´1if we define the notation νN´n´1“ν0as follows: ν0pΩc1¨¨¨cN´1cq:“1Ωpc1,...,cN´1, cq.(5.49) We will use this notation in the sequel. For pc1,...,cNq P CNfixed, we define decreasing sequences of sets tAc1¨¨¨cnuN n“1and tBc1¨¨¨cnuN n“1as follows: Ac1¨¨¨cn:“Ac1X...XAcnand Bc1¨¨¨cn:“Bc1X...XBcn,1ďnďN. The definition formally makes sense for pc1,...,cNq P CN, but will only be useful for pc1,...,cNq P Ω. Namely, if pc1,...,cNq P Ω, then it follows from the definition (5.46) that |Ac1¨¨¨cn| ě |Ac1¨¨¨cN| ě 1 2¨δNζ|A|and |Bc1¨¨¨cn| ě 1 2¨δNζ |B|.(5.50) We now construct the sets tHnuN n“1Ăδ¨Z. At the same time, we will construct subsets C1,...,CNĂC, and points cnPCn,1ďnďN, with the properties νN´npΩc1¨¨¨cnq ě 2´n´1δNζ and νpCnq ě 2´n´1δNζ,1ďnďN. (5.51) ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 49 In particular, the first part of (5.51) with n“Nshows that pc1,...,cNq P Ω, recall the notation (5.49). To begin with, we define C1:“ tcPC:νN´1pΩcq ě 2´2δNζ u, and we choose an arbitrary element c1PC1. Since żνN´1pΩcqdνpcq “ νNpΩq ě 2´1δNζ by (5.47), and the case n“0of (5.48), we observe that νpC1q ě 2´2δNζ by Chebyshev’s inequality. In particular C1‰ H. We then define H1:“ pc1Bc1qδ. Assume inductively that H1,...,Hnand C1,...,CnĂC, and cjPCj,1ďjďnď N´1, have already been constructed, and satisfy (5.51). We then pick an element cn`1P Cn`1, where Cn`1:“ tcPC:νN´n´1pΩc1¨¨¨cncq ě 2´n´2δNζ u,1ďnďN´1. For n“N´1, the notation νN´n´1pΩc1¨¨¨cncqshould be interpreted as in (5.49), so CN“ tcPC:1Ωpc1,...,cN´1, cq ě 2´N´1δNζ u “ tcPC:pc1,...,cN´1, cq P Ωu. For an arbitrary choice cn`1PCn`1, we note that the first part of (5.51) is satisfied with index "n`1", simply by the definition of Cn`1. The set Cn`1also satisfies the second part of (5.51) with index "n`1", by 2´n´1δNζ (5.51) ďνN´npΩc1¨¨¨cnq(5.48) “żνN´n´1pΩc1¨¨¨cncqdνpcq, and Chebyshev’s inequality. Whereas c1PC1was chosen arbitrarily, the element cn`1PCn`1is chosen in such a way that the quantity |Hn`cn`1Bc1¨¨¨cn`1|δis maximised, among all possible choices cn`1PCn`1. For this choice of cn`1PCn`1, we define Hn`1:“Hn`pcn`1Bc1¨¨¨cn`1qδ. Proceeding in this manner yields a sequence of sets H1,...,HN, and a distinguished sequence pc1,...,cNq P Ω, which we fix for the remainder of the argument. We record that if pc1,¨¨¨, cnq,1ďnďN´1, is an initial sequence of pc1,¨¨¨, cNq, then |Bc1¨¨¨cnc| ě 1 2δNζ|Bc1¨¨¨cn| ě δ¯ǫ|Bc1¨¨¨cn|, c PCn`1.(5.52) The second inequality simply follows from our choice of ζat (5.43). To see the first inequality, recall from the definition of cPCn`1that (in particular) Ωc1¨¨¨cnc‰ H (in the case n“N´1simply pc1,...,cn, cq P Ω). This means that there exists a sequence pc1 n`2,...,c1 Nq P CN´n´1such that pc1,...cn, c, c1 n`2,...,c1 Nq P Ω. Consequently, |Bc1¨¨¨cnc| ě |Bc1X¨¨¨BcnXBcXBc1 n`2X¨¨¨Bc1 N| ě 1 2δNζ |B| ě 1 2δNζ|Bc1¨¨¨cn| by the definition of Ω, see (5.46). Note that HnĂ pδ¨Zq X r0, nsfor all 1ďnďNby a straightforward induction, so |Hn| ď 2Nδ´1. Therefore, by the pigeonhole principle, there exists an nP t1,...,N ´1u such that |Hn`1| ď p2Nδ´1q1{pN´1q|Hn| ď 4δ´1{pN´1q|Hn|.(5.53) 50 TUOMAS ORPONEN We now consider the objects ¯ A:“Hn,¯ B:“Bc1¨¨¨cn,and ¯ν:“νpCn`1q´1ν|Cn`1.(5.54) We will show in a moment these objects satisfy the hypotheses of Theorem 5.3 with constants ¯α, ¯ β, ¯κ, ¯γ, and ¯ǫ0. First, however, we conclude the proof of Theorem 1.6, taking this for granted. By Theorem 5.3, there exists ¯cPCn`1(a set of full ¯νmeasure) such that whenever B1Ă¯ Bis a set of cardinality |B1| ě δ¯ǫ|B|, we have |Hn`¯cB1|δ“ | ¯ A`¯cB1|δěδ´¯ǫ|¯ A| “ δ´¯ǫ|Hn|.(5.55) (To be accurate, Theorem 5.3 only claims this for some ¯cPsptp¯νq, but the proof showed, see (5.10), that actually the set of non-admissible cPsptp¯νqhave measure strictly smaller than 1, so we can pick cPCn`1.) However, for every cPCn`1, the set B1:“Bc1¨¨¨cncĂ Bc1¨¨¨cn“¯ Bsatisfies |B1|(5.52) ěδ¯ǫ|¯ B|and |Hn`cB1|δ.|Hn`1|(5.53) ď4δ´1{pN´1q|Hn|(5.42) ďδ´¯ǫ{2|Hn|.(5.56) The inequality |Hn`cB1|δ.|Hn`1|follows from the fact that whenever cPCn`1, the set Hn` pcB1qδ“Hn` pcBc1¨¨¨cncqδis a competitor in the definition of Hn`1. With the choice c“¯cPCn`1, the inequalities (5.55)-(5.56) are mutually incompatible for δą0 small enough, depending on ¯ǫ“¯ǫpα, β, γ, κq ą 0. A contradiction has been reached. It remains to check that that the objects in (5.53) satisfy the hypotheses of Theorem 5.3 with constants ¯α, ¯ β, κ{2,¯γ, and ¯ǫ0. More precisely: (a) |¯ A| ď δ´¯α, (b) |¯ B| ě δ´¯ β, and ¯ Bsatisfies a Frostman condition with exponent ¯κ, for rP rδ, δ¯ǫ0s, (c) ¯νsatisfies a Frostman condition with exponent ¯γ. We first use the Plünnecke-Ruzsa inequality to establish (a), assuming that δą0is sufficiently small in terms of N, ¯α. It is clear by induction that Hncan be written as a sum of nďNsets of the form pcmBc1¨¨¨cmqδ, for some 1ďmďn. Noting that Ac1¨¨¨cnĂAcm, each of these sets individually satisfies |Ac1¨¨¨cn`pcmBc1¨¨¨cmqδ|.|Acm`cmBcm|δ (5.45) ďδ´ζ|A|(5.50) ď2δ´pN`1qζ|Ac1¨¨¨cn|. We may therefore infer that |Hn|.Nδ´NpN`1qζ|A| ď δ´NpN`1qζ´α. from the Plünnecke-Ruzsa inequality, Lemma 3.3, applied with Ac1¨¨¨cnin place of A(and finally also using |Ac1¨¨¨cn| ď |A| ď δ´α, see above (5.44)). This inequality implies |Hn| ď δ´¯αfor small enough δą0, recalling our choice of ζat (5.43). We move to (b). Recall from (5.44) that the set Bsatisfies the assumptions of Theorem 1.6 with constants ǫ0, κ ą0: |BXBpx, rq| ď rκ|B|, x PR, δ ďrďδǫ0“δ¯ǫ0. Since Bc1¨¨¨cnĂB, and |Bc1¨¨¨cn| ě 1 2δNζ |B|by (5.50), we deduce that Bc1¨¨¨cnsatisfies a Frostman condition with exponent ¯κ: |Bc1¨¨¨cnXBpx, rq| ď 2δ´Nζ rκ|Bc1¨¨¨cn| ď r¯κ|Bc1¨¨¨cn|, x PR, δ ďrďδ¯ǫ0. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 51 The final inequality uses our choice of ζin (5.43), and also assumes that δą0is sufficiently small, depending on ¯ǫ0, κ. Moreover, since |B| ě δ´βby assumption, we have |Bc1¨¨¨cn| ě 1 2δNζ |B|(5.43) ěδ´¯ β. Let us finally check (c), namely that the probability measure ¯ν“νpCn`1q´1¨ν|Cn`1satisfies a Frostman condition with exponent ¯γ. Indeed, recalling from (5.51) that νpCn`1q ě 2´n´2δNζ , we have ¯νpBpx, rqq ď 2n`2δ´NζνpBpx, rqq ď 2N`2δ´Nζ ¨rγ, x PR, δ ďrďδ¯ǫ0. Since rγďδǫ0pγ´¯γqr¯γfor rďδǫ0, by our choice of ζin (5.43), the right hand side is bounded from above by r¯γfor all δą0small enough, depending on N, γ, ¯γ(all of which only depend on α, β, γ, κ). We have now verified that the objects ¯ A, ¯ B, ¯νfrom (5.54) satisfy the hypotheses of Theorem 5.3. This concludes the proof of Theorem 1.6. Remark 5.57.Once again, in order to deduce Theorem 1.6 for a fixed exponent "κ" from Theorem 5.3, we only needed to apply Theorem 5.3 with a fixed exponent ¯κP p0, κq, as close to κas we desire. Combining this with the previous similar Remarks 5.11-5.37, we obtain the conclusion alluded to in Remark 1.9: to deduce Theorem 1.6 for a fixed exponent "κ" from Theorem 1.8, we only needed to apply Theorem 1.8 for ¯κP p0, κq arbitrarily close to κ. 5.6. Proof of Corollary 1.7.I close the paper by recording the (standard pigeonholing) proof of Corollary 1.7, whose statement is recalled here: Corollary 5.58. Let 0ăβďαă1and κą0. Then, there exists η“ηpα, β, κq ą 0such that if A, B ĂRare Borel sets with dimHA“α,dimHB“β, then dimHtcPR: dimHpA`cBq ď α`ηu ď α´β 1´β`κ. Proof. It is easy to reduce to the case where A, B are compact, A, B Ă r0,1s, and HαpAq ą 0and HβpBq ą 0. In this case, one may use Frostman’s lemma [23, Theorem 8.8] to find Borel probability measures µA, µBwith sptpµAq Ă A,sptpµBq Ă B, and satisfying µApBpx, rqq ď CArαand µBpBpx, rqq ď CBrβfor all balls Bpx, rq Ă R. If ηą0is small enough, we will show that dimHEď pα´βq{p1´βq`κ, where E:“Eη:“ tcP r1 2,1s: dimHpA`cBq ă α`ηu. It is easy to show (by rescaling considerations) that this implies Corollary 1.7, where r1 2,1sis replaced by R. It is well-known that the set EĂ r1 2,1sis Borel. Consequently, if the inequality fails, one may use Frostman’s lemma again to find a Borel probability measure ν, supported on E, satisfying νpBpx, rqq ď Cνrγfor all xPRand rą0, where γě pα´βq{p1´βq`κ. For future reference, we fix some parameters ¯αąα,¯ βăβ, and ¯γăγsuch that the inequality ¯γą p¯α´¯ βq{p1´¯ βq(5.59) still holds. We then let ¯ǫ, ¯ǫ0,¯ δ0ą0be the constants provided by Theorem 1.6 applied with parameters ¯α, ¯ β, κ “¯ β, ¯γ. We pick ηą0in the definition of Eso small that ηămint¯ǫ, ¯α´αu.(5.60) 52 TUOMAS ORPONEN Fix cPsptpνq Ă E, so dimHpA`cBq ă α`η. This means that for a given fixed threshold δ0:“2´j0P2´N(the requirements will depend on α, β, γ, CA, CB, Cν), one may find a countable cover Icof A`cB, consisting of disjoint dyadic intervals of length ℓpIq ď δ0, such that ÿ IPIc ℓpIqα`ηď1.(5.61) Below, we will often write that something holds "for small enough δą0": this will always mean "assuming that the upper bound δ0for δhas been chosen sufficiently small, depending on the parameters α, β, γ, CA, CB, Cν. In particular, we will take δ0ď¯ δ0. The "tubes" Tc:“ tπ´1 cpIquIPIccover AˆBĄsptpµAˆµBq, so żEÿ TPTcpµAˆµBqpTqdνpcq “ 1. Recall that δ0“2´j0, and let Ij c:“ tIPIc:ℓpIq “ 2´jufor jěj0. Write also Tj c:“ tπ´1 cpIquIPIj c. Since Tc“Ťjěj0Tj c, there exists jěj0such that żEÿ TPTj c pµAˆµBqpTqdνpcq&j´2. Write δ:“2´jfor this index j. According to the estimate above, there exists a subset E1 δĂEof measure νpE1 δq&j´2“log2p1{δq´2such that for each cPE1 δ, the tubes TPTj c cover a subset GcĂsptpµAˆµBqof measure pµAˆµBqpGcq&log2p1{δq´2. In particular, we record that |πcpGcq|δď |Tj c| ď δ´α´η, c PE1 δ,(5.62) by (5.61). For the remainder of this argument, we use the notation f/gto abbreviate an inequality of the form fďClog2p1{δqCgfor some constant Cą0, which may depend on the Frostman constants α, β, γ, CA, CB, Cν. In particular, j´2“log2p1{δq´2'1. For xPR, let Iδpxq P Dδbe the unique dyadic interval of length δwith xPIδpxq. We now split the set Aas follows: A“ď ρP2´N Apρq:“ txPA:ρďµApIδpxqq ă 2ρu. We define the sets Bpρq Ă Bsimilarly. Since µApIδpxqq ď CAδαand µBpIδpyqq ď CBδβ, we see that Apρq ‰ H implies ρďCAδα, and Bpρq ‰ H implies ρďCβδβ. We also note that Apρqcan be expressed as the intersection of Awith certain dyadic intervals Apρq Ă Dδ. The same is true for Bpρq, for certain dyadic intervals Bpρq Ă Dδ. Let µApρqbe the restriction of µAto the intervals Apρq, and similarly let µBpρqbe the restriction of µBto the intervals in Bpρq. Then ÿ ρ1ÿ ρ2żE1 δpµApρ1qˆµBpρ2qqpGcq « 1,(5.63) so it follows from the pigeonhole principle that żE1 δpµApρAqˆµBpρAqqpGcq « 1 ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 53 for some fixed choices ρAďCAδαand ρBďCBδβ(noting that values ρ1, ρ2ďδ2cannot contribute substantially to the sum in (5.63)). In particular, there exists a further subset EδĂE1 δwith the property pµApρAqˆµBpρBqqpGcq « 1for all cPEδ. We now abbreviate ¯µA:“µApρAqand ¯µB:“µBpρBq, so }¯µA} « 1« }¯µB}. The measure ¯µAis supported on the closure of the intervals in ApρAq, and ¯µBis supported on the closure of the intervals in BpρBq. Let Aδ:“ pδ¨ZqXpYApρAqq and Bδ:“ pδ¨ZqXpYBpρBqq. We observe that ρA¨|Aδ| „ }µA} « 1ùñ ρA« |Aδ|´1,(5.64) and similarly ρB« |Bδ|´1. Since ρAďCAδα, we record that |Aδ| « ρ´1 A'δ´α.(5.65) We next claim that, somewhat conversely, |Aδ| ď δ´¯αif δą0is sufficiently small. To see this, fix an arbitrary cPEδ. Since p¯µAˆ¯µBqpGcq « 1, there exists bPsptp¯µBqsuch that ¯µApGcpbqq « 1,where Gcpbq “ txPsptp¯µAq:px, bq P Gcu. Now, if Gcpbq:“ tIPApρAq:Gcpbq X I‰ Hu, we see that ¯µApIq „ ρAfor all IPGcpbq, and ¯µApYGcpbqq ě ¯µApGcpbqq « 1. Moreover, we observe that |Gcpbq|δ.|πcpGcq|δ, since πcpGcq Ą Gcpbq`bc. Putting these observations together, |Aδ|(5.64) «ρ´1 A/ρ´1 A¨¯µApYGcpbqq .|Gcpbq|δ.|πcpGcq|δ (5.62) ďδ´α´η.(5.66) Since α`ηă¯αby (5.60), the inequality |Aδ| ď δ´¯αholds for δą0sufficiently small. Next, since ρBďCBδβ, we record that |Bδ| « ρ´1 B'δ´βùñ |Bδ| ě δ¯ β,(5.67) where the implication holds if δą0is sufficiently small. Moreover, for xPRand rěδ, we note that every point yPBδXBpx, rqis contained in an interval Iypδq P BpρBqwith µBpIypδqq ě ρB. Since Iypδq Ă Bpx, 2rq, we deduce that |BδXBpx, rq| ď ρ´1 B¨µBpBpx, 2rqq ď ρ´1 B¨CBp2rqβ/rβ|Bδ|.(5.68) In particular, for the parameter ¯ǫ0ą0fixed below (5.59), we have |BδXBpx, rq| ď r¯ β|Bδ| for δďrďδ¯ǫ0, provided that δą0is small enough. Finally, the measure νδ:“νpEδq´1¨ν|Eδsatisfies νδpBpx, rqq /νpBpx, rqq ď Cνrγ, r ą0,(5.69) so the inequality νδpBpx, rqq ď r´¯γholds for all rďδ¯ǫ0, provided that δą0is small enough. The estimates (5.66)-(5.69), and (5.59), imply that the triple Aδ, Bδ, νδsatisfies all the hypotheses of Theorem 1.6 with constants ¯α, ¯ β, κ “¯ β, ¯γ, and ¯ǫ0. Consequently, there exists cPEδĂE1 δ(a set of full νδmeasure) such that |πcpGq|δěδ´¯ǫ|Aδ| (5.65) 'δ´α´¯ǫ(5.70) for all subsets GĂAδˆBδof cardinality |G| ě δ¯ǫ|A||B|. We argue that this contradicts (5.62). The only issue is that set GcĂsptpµAˆµBqis not exactly a subset of AδˆBδ. To fix this, recall that nevertheless p¯µAˆ¯µBqpGcq « 1. Let Gc:“ tIˆJPApρAqˆBpρBq:pIˆJqXGc‰ Hu. 54 TUOMAS ORPONEN Then Gcis a cover of Gc, and p¯µAˆ¯µBqpQq „ ρAρB« |Aδ|´1|Bδ|´1for all Q“IˆJPGc. Consequently, |Gc|&pρAρBq´1¨p¯µAˆ¯µBqpGcq « |Aδ||Bδ|. Now, let Gc,δ Ă pAδˆBδqXGcp2δqbe subset of cardinality |Gc,δ|'|Aδ||Bδ|. In particular |Gc,δ| ě δ¯ǫ|Aδ||Bδ|for δą0small enough. Therefore the estimate (5.70) holds for G“ Gc,δ. On the other hand, since Gc,δ ĂGcp2δq, we have |πcpGc,δq|δ.|πcpGcq|δďδ´α´η by (5.62). Since we chose ηă¯ǫin (5.60), this estimate is not compatible with (5.70). A contradiction has been reached, and the proof of Corollary 1.7 is complete.  REFERENCES [1] Yves Benoist and Nicolas de Saxcé. A spectral gap theorem in simple Lie groups. Invent. Math., 205(2):337–361, 2016. [2] J. Bourgain. On the Erdös-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal., 13(2):334–365, 2003. [3] J. Bourgain and A. Gamburd. A spectral gap theorem in SUpdq.J. Eur. Math. Soc. (JEMS), 14(5):1455– 1511, 2012. [4] Jean Bourgain. Multilinear exponential sums in prime fields under optimal entropy condition on the sources. Geom. Funct. Anal., 18(5):1477–1502, 2009. [5] Jean Bourgain. The discretized sum-product and projection theorems. J. Anal. Math., 112:193–236, 2010. [6] Jean Bourgain and Alex Gamburd. On the spectral gap for finitely-generated subgroups of SUp2q.Invent. Math., 171(1):83–121, 2008. [7] Damian D ˛abrowski, Tuomas Orponen, and Michele Villa. Integrability of orthogonal projections, and applications to Furstenberg sets. Adv. Math., 407:Paper No. 108567, 34, 2022. [8] P. Erd˝os and E. Szemerédi. On sums and products of integers. In Studies in pure mathematics, pages 213–218. Birkhäuser, Basel, 1983. [9] K. J. Falconer. Hausdorff dimension and the exceptional set of projections. Mathematika, 29(1):109–115, 1982. [10] Yuqiu Fu, Shengwen Gan, and Kevin Ren. An incidence estimate and a Furstenberg type estimate for tubes in R2.J. Fourier Anal. Appl., 28(4):Paper No. 59, 28, 2022. [11] M. Z. Garaev. An explicit sum-product estimate in Fp.Int. Math. Res. Not. IMRN, (11):Art. ID rnm035, 11, 2007. [12] A. A. Glibichuk and S. V. Konyagin. Additive properties of product sets in fields of prime order. In Additive combinatorics, volume 43 of CRM Proc. Lecture Notes, pages 279–286. Amer. Math. Soc., Providence, RI, 2007. [13] Larry Guth, Nets Hawk Katz, and Joshua Zahl. On the discretized sum-product problem. Int. Math. Res. Not. IMRN, (13):9769–9785, 2021. [14] Larry Guth, Noam Solomon, and Hong Wang. Incidence estimates for well spaced tubes. Geom. Funct. Anal., 29(6):1844–1863, 2019. [15] Katalin Gyarmati, Máté Matolcsi, and Imre Z. Ruzsa. Plünnecke’s inequality for different summands. In Building bridges, volume 19 of Bolyai Soc. Math. Stud., pages 309–320. Springer, Berlin, 2008. [16] Weikun He. Discretized sum-product estimates in matrix algebras. J. Anal. Math., 139(2):637–676, 2019. [17] Weikun He. Orthogonal projections of discretized sets. J. Fractal Geom., 7(3):271–317, 2020. [18] Weikun He and Nicolas de Saxcé. Sum-product for real Lie groups. J. Eur. Math. Soc. (JEMS), 23(6):2127– 2151, 2021. [19] Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2), 180(2):773–822, 2014. [20] Robert Kaufman. On Hausdorff dimension of projections. Mathematika, 15:153–155, 1968. [21] Jialun Li. Discretized Sum-product and Fourier decay in Rn.J. Anal. Math., 143(2):763–800, 2021. [22] J. M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3), 4:257–302, 1954. ON THE DISCRETISED ABC SUM-PRODUCT PROBLEM 55 [23] P. Mattila. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. 1st paperback ed. Cambridge: Cambridge University Press, 1st paperback ed. edition, 1999. [24] Ali Mohammadi and Sophie Stevens. Attaining the exponent 5/4 for the sum-product problem in finite fields. Int. Math. Res. Not. IMRN, (4):3516–3532, 2023. [25] Daniel M. Oberlin. Some toy Furstenberg sets and projections of the four-corner Cantor set. Proc. Amer. Math. Soc., 142(4):1209–1215, 2014. [26] Tuomas Orponen. On the distance sets of Ahlfors-David regular sets. Adv. Math., 307:1029–1045, 2017. [27] Tuomas Orponen. On arithmetic sums of Ahlfors-regular sets. Geom. Funct. Anal., 32(1):81–134, 2022. [28] Tuomas Orponen and Pablo Shmerkin. On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. Duke Math. J. (to appear). [29] Tuomas Orponen and Laura Venieri. A note on expansion in prime fields. arXiv e-prints, page arXiv:1801.09591, January 2018. [30] Yuval Peres and Wilhelm Schlag. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J., 102(2):193–251, 2000. [31] Orit E. Raz and Joshua Zahl. On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem. Geom. Funct. Anal. (to appear). [32] Misha Rudnev and Sophie Stevens. An update on the sum-product problem. Math. Proc. Cambridge Philos. Soc., 173(2):411–430, 2022. [33] Imre Z. Ruzsa. An application of graph theory to additive number theory. Sci. Ser. A Math. Sci. (N.S.), 3:97–109, 1989. [34] Pablo Shmerkin. On Furstenberg’s intersection conjecture, self-similar measures, and the Lqnorms of convolutions. Ann. of Math. (2), 189(2):319–391, 2019. [35] Pablo Shmerkin. On the Hausdorff dimension of pinned distance sets. Israel J. Math., 230(2):949–972, 2019. [36] Pablo Shmerkin. A nonlinear version of bourgain’s projection theorem. (J. Eur. Math. Soc. to appear), 2020. [37] Pablo Shmerkin and Hong Wang. On the distance sets spanned by sets of dimension d{2in Rd.arXiv e-prints, page arXiv:2112.09044, December 2021. [38] Sophie Stevens and Frank de Zeeuw. An improved point-line incidence bound over arbitrary fields. Bull. Lond. Math. Soc., 49(5):842–858, 2017. [39] Endre Szemerédi and William T. Trotter, Jr. Extremal problems in discrete geometry. Combinatorica, 3(34):381–392, 1983. [40] Terence Tao and Van Vu. Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF JYVÄSKYLÄ, P.O. BOX 35 (MAD), FI-40014 UNIVERSITY OF JYVÄSKYLÄ, FINLAND Email address:[email protected]