Some new thin sets of integers in harmonic analysis

arXiv:0912.4214v1 [math.FA] 21 Dec 2009 Some new thin sets of integers in Harmoni Analysis Daniel Li, Hervé Queéle, Luis Rodríguez-Piazza Abstrat . We randomly onstrut various subsets Λ of the integers whih have both smal lness and largeness properties. They are smal l sine they are very lose, in various meanings, to Sidon sets: the ontinuous funtions with spetrum in Λ have uniformly onvergent series, and their Fourier oeients are in ℓp for al l p > 1 ; moreover, al l the Lebesgue spaes Lq Λ are equal for q < +∞ . On the other hand, they are large in the sense that they are dense in the Bohr group and that the spae of the bounded funtions with spetrum in Λ is non separable. So these sets are very dierent from the thin sets of integers previously known. Résumé . On onstruit aléatoirement des ensembles Λ d'entiers positifs jouissant simultanément de propriétés qui les font apparaître à la fois omme petits et omme grands. Ils sont petits ar très prohes à plus d'un égard des ensembles de Sidon: les fontions ontinues à spetre dans Λ ont une série de Fourier uniformément onvergente, et ont des oeients de Fourier dans ℓp pour tout p > 1 ; de plus, tous les espaes de Lebesgue Lq Λ oïnident pour q < +∞ . Mais ils sont par ail leurs grands au sens où ils sont denses dans le ompatié de Bohr et où l'espae des fontions bornées à spetre dans Λ n'est pas séparable. Ces ensembles sont don très diérents des ensembles mines d'entiers onnus auparavant. Key-words . ergo di set  launary set  Λ(q) -set  quasi-indep endent set  random set  p -Rider set  Rosenthal set  p -Sidon set  set of uniform onvergene  uniformly distributed set. Mathematis Sub jet Classiation . Primary : 42A36  42A44  42A55  42A61  43A46; Seondary : 60D05 Intro dution It is well known that the Fourier series of an integrable funtion dened on the unit-irle T=R/2πZ of the omplex plane C an b e badly b ehaved. For example, it is well known that there exist ontinuous funtions whose Fourier series is not everywhere onvergent (see [30℄, Th. 18.1, and Th. 19.5 for the optimal result), and integrable ones with everywhere divergent Fourier series (see [30℄, Th. 19.2 for instane; see also [29℄). The problem of thin sets of integers is the following: instead of onsidering all the integrable funtions on T , or all the ontinuous ones, we onsider only 1 those whose sp etrum (the set where their Fourier o eients do not vanish) is ontained in a presrib ed subset Λ of the integers Z . This set Λ will b e said thin if the Fourier series of these funtions b ehaves b etter than in the general ase. A typial example is Λ = {1,3,32,...,3n,...} . It is well known (see [62℄, for instane) that every integrable funtion f with sp etrum in Λ ( f∈L1 Λ ) is atually square integrable, and that every ontinuous funtion f with sp etrum in Λ ( f∈ CΛ ) has a normally onvergent Fourier series (equivalently b f∈ℓ1 ). In his seminal pap er [54℄, W. Rudin dened two notions of thinness for Λ : Λ is a Sidon set if f∈ CΛ implies that b f∈ℓ1 , and Λ is a Λ(q) -set for some q > 1 , if f∈L1 Λ implies that f∈Lq . These onepts may as well b e dened in the more general setting of a ompat ab elian group G equipp ed with its normalized Haar measure, and for a subset Λ of its disrete dual group Γ . W. Rudin studied the general prop erties of those sets and the onnetion b etween the two notions. In partiular, he showed that Sidon sets are Λ(q) -sets for all q < +∞ , and that, more preisely: (0.1) Λ Sidon implies kfkq≤C√qkfk2 for every Λ -p olynomial f and for every q≥2 , where C is a onstant whih depends only on the Sidon onstant of Λ . Sine then, several new notions of thin sets emerged. These inlude p -Sidon sets (see [2℄, [3℄, [4℄, [5℄, [14℄, [18℄, [21℄, [23℄, [24℄, [34℄, [47℄, [60℄), and sets of uniform onvergene (see [1℄, [19℄, [20℄, [21℄, [26℄, [33℄, [44℄, [45℄, [56℄, [57℄): every ontinuous funtion with spetrum in suh a set has its Fourier seriesin ℓp or uniformly onvergent, resp etively. But the examples of suh sets were always nearly the same: pro duts (sometimes frational pro duts: [3℄, [4℄, [5℄), or sums of Sidon sets, whih is a severe restrition for the geometry of the Banah spae CΛ . For example, F. LustPiquard ([40℄) proved that: (0.2) The injetive tensor pro dut ℓ1b ⊗ε···b ⊗εℓ1 has the Shur prop erty ( i.e. weakly null sequenes onverge in norm to zero). It follows easily that: (0.3) If Λ = E1×···×Ek , where the Ej 's are Sidon sets, then CΛ has the Shur prop erty; in partiular, CΛ do es not ontain c0 , the spae of sequenes going to zero at innity. Sine these sets were essentially the only known examples of p -Sidon sets (they are exatly 2N/(N+ 1) -Sidon), one ould b elieve that all p -Sidon sets have this prop erty. It should b e mentionned that in [3℄, R. Blei onstruted for eah p∈]1,2[ , exatly p -Sidon sets, using frational pro duts, so of a dierent typ e, but the orresp onding spae CΛ app ears as an ℓ1 -sum of nite dimensional spaes, and so do es have the Shur prop erty (we thank R. Blei for this remark). Beause of this lak of examples, the omparison between two lasses of thin sets proved to be very diult: whether a p -Sidon, or a set of uniform onvergene is a Λ(q) -set for some q > 1 is still an op en problem. On the other 2 hand, onsiderable progress onerning the Sidon sets or Λ(q) -sets has b een made: for example, G. Pisier ([47℄, Th. 6.2) proved that the onverse of (0.1) is true, and J. Bourgain ([12℄) proved that for eah q > 2 there exist exatly Λ(q) -sets, i.e. sets whih are Λ(q) , but Λ(q′) for no q′> q . Both authors used random metho ds, and more sp eially, J. Bourgain p opularized the metho d of seletors to pro due several thin sets Λ with unusual prop erties, suh as b eing uniformly distributed, whih implies, by a result of F. LustPiquard ([42℄), that CΛ ontains c0 and therefore is not a Rosenthal set ( i.e. there are b ounded measurable funtions with sp etrum in Λ wih are not almost everywhere equal to a ontinuous funtion), and whih also implies that Λ is dense in the Bohr group (see [6℄, Theorem 1). This allowed the rst named author to see that there are sets of integers whih are Λ(q) for all q < +∞ but not Rosenthal ([37℄; see also [43℄). The aim of this pap er is the onstrution of random sets Λ of integers whih have thinness prop erties, but wih are not Rosenthal sets ( i.e. CΛ is not the whole L∞ Λ ), atually suh that CΛ ontains c0 , and are dense in the Bohr group. In view of (0.3), these sets will neessarily b e very exoti ompared to the previously known examples. This shows that replaing absolute onvergene of the Fourier series by uniform onvergene (sets of uniform onvergene) or by ℓp onvergene for p > 1 ( p -Sidon sets) gives sets whih are very far from Sidon sets. This onstrasts with Pisier's result saying that Λ is neessarily a Sidon set whenever b f∈ℓ1,∞ for every f∈ CΛ (from [48℄, Théorème 2.3 (vi), and the top of page 688). On the other hand, though non-Sidon Rosenthal sets do exist ([53℄), it follows from Bourgain-Milman's otype theorem ([13℄) that, for every non-Sidon set Λ , CΛ do es ontain ℓn ∞ uniformly, so that the presene of c0 inside CΛ for non-Sidon Λ may app ear not so surprising. Although it is not known whether Sidon sets may be dense in the Bohr group, we obtain in this paper, as mentioned ab ove, sets whih are dense in the Bohr group, and are of uniform onvergene and p -Sidon for every p > 1 . We onstrut essentially four types of sets. Eah of them will b e a non Rosenthal set, but a set of uniform onvergene, Λ(q) for all q < +∞ , and with moreover additional prop erties of p -Sidoniity. The rst one (Theorem 2.2) is a very launary set Λ with the niest properties: it is p -Sidon for all p > 1 . The seond and third ones (Theorem 2.5 and Theorem 2.6) are medium launary sets: for eah p with 1< p < 4/3 , they are, in Theorem 2.5, p -Rider (a weaker prop erty than b eing p -Sidon, see the denition b elow), but not q -Rider for q < p , and are q -Sidon for every q > p/(2 −p) ; and in Theorem 2.6, they are q -Rider for every q > p , but not p -Rider, and they are q -Sidon for every q > p/(2 −p) . Finally, the fourth type (Theorem 2.7) is a set Λ whih is, in some sense as little launary as possible if we want its trae on eah interval [N, 2N[ to have a b ounded Sidon onstant. It leads to sets whih are 4/3-Rider, but not q -Rider for q < 4/3 . We onstrut these sets by using various hoies of seletors, and adding arithmetial, funtional or probabilisti arguments. The treatment of the last ase requires a dierent probabilisti approah, taken from [8℄. 3 It should b e noted that in the two rst ases the sets are uniformly distributed; in the fourth ase , however, the sets Λ only have p ositive upp er density in uniformly distributed sets. Nevertheless, CΛ still ontains c0 , by a result of F. Lust-Piquard ([42℄, Th. 5). Aknowledgement. Part of this pap er was made when the rst named author was a guest of the Departamento de Análisis Matemátio de la Universidad de Sevilla in April 1999, and when the third named author was a guest of the Université d'Artois in Lens in june 1999. 1 Notation, denitions and preliminary results We denote by T the ompat ab elian group of omplex numb ers of mo dulus one, equipp ed with its normalized Haar measure m . C(T) denotes the spae of ontinuous omplex funtions dened on T , equipp ed with its sup norm kk∞ and identied as usual with the spae of ontinuous 2π -p erio di omplex funtions dened on R . If Λ is a subset of the dual group Z , CΛ will denote the subspae of C(T) onsisting of funtions whose sp etrum lies in Λ : b f(n)≡ZT fe−ndm = 0 if n∈Z\Λ, where en(z) = zn , or equivalently, en(t) = eint . CΛ is the uniform losure of the spae PΛ of trigonometri p olynomials with sp etrum in Λ , i.e. the uniform losure of the subspae PΛ generated by the haraters en , with n∈Λ . For f∈ C(T) , 1≤q < +∞ , M and N p ositive integers, we shall denote the Fourier sums of f by: SM,N (f) = N X −Mb f(n)en and the symmetri Fourier sums of f by: SN(f) = SN,N (f) = N X −Nb f(n)en. |A| denotes the ardinality of the nite set A . A relation in Λ⊆Z∗≡Z\ {0} is a (+1,−1,0) -valued sequene (θk)k∈Λ suh that P|θk|<+∞ and Pθkk= 0 . The set S={k;θk6= 0} is alled the support of the the relation, and |S|=P|θk| is alled its length . The relation (θ′ k)k∈Λ is said to b e longer than the relation (θk)k∈Λ if θk6= 0 implies θk=θ′ k . The set Λ⊆Z∗ is quasi-independent if it ontains no non-trivial relation ( i.e. with non-empty supp ort). Typially, Λ = {1,2,4,...,2n,...} is quasiindep endent. The quasi-indep endent sets are the prototype of Sidon sets, i.e. 4 of sets Λ for whih: kb fk1≤Kkfk∞ for all f∈ CΛ . The b est onstant K in this inequality is alled the Sidon onstant of Λ and is denoted by S(Λ) . We will refer to [39℄ for standard notions on Sidon sets. It is known that quasi-independent sets are not only Sidon sets but their Sidon onstant is bounded by an absolute onstant: this follows from [54℄, Th. 2.4 and [49℄, Lemma 1.7. Other pro ofs an b e found in [48℄, lemme 3.2, and in [9℄, Prop. 1. We shall use the fat that S(Λ) ≤8 if Λ is quasi-indep endent. Let us reall now some lassial denitions and results. A set Λ⊆Z is said to b e a Λ(q) -set (where q > 2 ) if there exists a p ositive onstant Cq suh that kfkq≤Cqkfk2 for every f∈ PΛ . The notion of a Λ(q) -set is, in some sense, lo al. That follows from the Littlewo o d-Paley theory. The next prop osition is essentially well-known, exept for the growth of the onstant, for whih we have found no referene. Aordingly, we oer a short pro of. Prop osition 1.1 Let Λ⊆[2,+∞[ . Then: (a) Let (Mn)n≥1 be a sequene of positive integers suh that M1≤2 and Mn+1/Mn≥α > 1 . If Λ∩[Mn, Mn+1[ , n≥1 , has a uniformly bounded Sidon onstant, then Λ is Λ(q) for al l q≥2 ; more preisely: kfkq≤C(q, α)kfk2 for every f∈ PΛ . (b) If Λ∩[2n,2n+1[ , n≥1 , has a uniformly bounded Sidon onstant, Λ is Λ(q) for every q≥2 and, more preisely: kfkq≤Cq2kfk2 for every f∈ PΛ and for some numerial onstant C . Pro of. (a) Set fk=X Mk≤n<Mk+1 b f(n)en and Sf =+∞ X k=1 |fk|21/2. Sine Mk+1/Mk≥α > 1 and Λ⊆[M1,+∞[ , we have ([62℄, Chap. XV, Th. 2.1): kfkq≤C0(q, α)kSfkq. Now, using the 2-onvexity of the Lq -norm for q≥2 , we obtain: kSfkq≤+∞ X k=1 kfkk2 q1/2. But fk∈ PΛk , where Λk= Λ ∩[Mk, Mk+1[ has a uniformly b ounded Sidon onstant. Therefore kfkkq≤C1√qkfk2 , where C1 is a numerial onstant. The result follows. (b) We now make use of the lassial square funtion Sg =X k∈Z|gk|21/2, 5 where gk=X 2k≤n<2k+1 bg(n)en,if k≥0 and gk=X −2|k|+1<n≤−2|k|bg(n)enif k < 0. For this lassial square funtion, we have the following sharp inequality, due to J. Bourgain ([11℄, Th. 1): kSgkp≤C0(p−1)−3/2kgkpfor 1 < p ≤2, where C0 is a numerial onstant. We dedue by duality that: kfkq≤C0q3/2kSfkqfor 2 ≤q < +∞. In fat, by orthogonality (reall that f∈ PΛ and that Λ⊆[2,+∞[ ) and the CauhyShwarz inequality, we have, for every g∈Lp with kgkp= 1 ( 1/p + 1/q = 1 ): |< f, g > |= +∞ X k=1 < fk, gk>=ZT +∞ X k=1 fk(−t)gk(t)dm(t) ≤ZT Sf(−t)Sg(t)dm(t) ≤ kSfkqkSgkp≤C0(p−1)−3/2kSfkq ≤C0q3/2kSfkq. This means that here we are allowed to take C0(q, 2) = C0q3/2 in part (a) of the pro of. The rest is unhanged, and we an also take C(q, 2) = C1√qC0q3/2= Cq2 .  A set Λ⊆Z is alled a set of uniform onvergene (in short a UC-set ) if, for any f∈ CΛ , the symmetri Fourier sums SN(f) onverge uniformly to f . Its onstant of uniform onvergene U(Λ) is the smallest onstant K suh that, for any f∈ CΛ : sup NkSN(f)k∞≤Kkfk∞. The following variant turns out to b e more tratable ([56℄). Λ is alled a set of omplete uniform onvergene (in short a CUC-set) if the translates ( Λ+a ) are uniformly UC for a∈Z , or equivalently, if the Fourier sums SM,N (f) onverge uniformly to f as M, N go to +∞ , for every f∈ CΛ . The two notions turn out to b e distint ([20℄), but learly oinide if Λ⊆N , whih will always be the ase in the sequel. The notion of CUC-set is also a lo al one as the following prop osition shows. Prop osition 1.2 ([57℄, Th. 3) Let Λ⊆N∗ and ΛN= Λ ∩[N, 2N[ . (a) If U(ΛN) is bounded by K for N= 1,2,... , then Λ is a CUC -set. (b) Let (Mn)n≥1 be a sequene of positive integers suh that Mn+1/Mn≥2 . Then, if Λ∩[Mn, Mn+1[ are quasi-independent for eah n , or more general ly if they are Sidon sets with uniformly bounded Sidon onstant, then Λ is a CUC -set. 6 Remark. (b) is a useful riterion to pro due sets that are CUC but not Sidon; for instane, if Λ = S+∞ n=1{2n+ 2j;j= 0,...,n−1} , then Λ∩[2n,2n+1[ is quasi-indep endent, whereas Λ∩[1, N] has about (log N)2 elements, and therefore annot b e Sidon (the mesh ondition for Sidon sets, see Prop osition 1.6 b elow, is violated). The random variables whih we shall use will always be dened on some probability spae (Ω,A,P) whih will play no expliit role, and the exp etation with resp et to P will always b e denoted by E : E(X) = ZΩ X(ω)dP(ω). Reall the (more or less) lassial deviation inequality (see [32℄,  6.3): Lemma 1.3 Let X1,...,XN be independent entered omplex random variables suh that |Xk| ≤ 1 , k= 1,...,N . Let σ≥ N P k=1 E|Xk|2 . Then, one has, for every a≤σ : P(|X1+···+XN| ≥ a)≤4 exp(−a2/8σ). Let (rn)n b e a Bernoulli sequene, i.e. a sequene of indep endent random variables suh that: P(rn= 1) = P(rn=−1) = 1/2. For f∈ P , the spae of trigonometri p olynomials, [[f]] denotes the norm of f in the Pisier's spae C a.s. : [[f]] = E X n rnb f(n)en ∞. See [25℄ and [47℄ for more information ab out this norm. Denition 1.4 A set Λ⊆Z is al led a p -Sidon set ( 1≤p < 2 ) if there exists a onstant K suh that kb fkp≤Kkfk∞ for al l f∈ PΛ . It is said to be a p -Rider set if there exists a onstant K suh that kb fkp≤ K[[f]] for al l f∈ PΛ . p -Rider sets were impliitely introdued, with dierent denition, in [18℄ (Th. 2.4), and in [23℄, p. 213, as lass Tp (see also [47℄, Th. 6.3). They were expliitely dened and studied in [51℄ and [52℄ under the name  p -Sidon presque sûrs. We used almost surely p -Sidon set in the rst version of this pap er, but, following a suggestion of J.-P. Kahane, we now use the terminology  p -Rider. Clearly, every p -Sidon set is p -Rider. The onverse is true for p= 1 : this is a remarkable result due to D. Rider ([50℄), making lever use of Drury's onvolution devie (whih proves that the union of two Sidon sets is Sidon [17℄). Whether this onverse is still true for 1< p < 2 is an op en problem. 7 Denition 1.5 We shal l say that a nite set B⊆Λ is M -pseudo-omplemented in Λ if there exists a measure µ on T suh that: |bµ| ≥ 1 on B;bµ= 0 on Λ\B;kµk ≤ M . The following prop osition gives some neessary, suient, or neessary and suient onditions for a set Λ to b e p -Sidon or p -Rider. Part (b) of this prop osition seems to b e new. Prop osition 1.6 Let Λ⊆Z∗ and 1≤p < 2 . Set ε(p) = 2/p −1 . Then: (a) Λ is a p -Rider set if and only if there exists a onstant δ > 0 suh that, for every nite set A⊆Λ , there exists a quasi-independent subset B⊆A suh that |B| ≥ δ|A|ε(p) . (b) Let q0>1 . If there exists a onstant δ > 0 suh that, for every nite set A⊆Λ , there exists a quasi-independent subset B⊆A suh that |B| ≥ δ|A|1/q0 and if B an moreover be taken M -pseudo-omplemented in Λ , for some xed M , then Λ is a q -Sidon set for every q > q0 . () If Λ is a p -Rider set, we have the fol lowing mesh ondition: |Λ∩[1, N]| ≤ C(log N)p/(2−p). Pro of. We refer to [51℄ for the pro of of (a) and (). To prove (b), let f∈ PΛ , x t > 0 , and set A={|b f|> t } . Take B⊆A and µ as in Denition 1.4. Then B is a Sidon set with Sidon onstant ≤8 , and sine f∗µ=P n∈Bb f(n)bµ(n)en , kfk∞≥M−1  X Bb f(n)bµ(n)en  ∞≥1 8MX B|b f(n)||bµ(n)| ≥1 8MX B|b f(n)| ≥ t|B| 8M≥tδ|A|1/q0 8M· In other words, for some onstant C > 0 , one has: t .|{|b f|> t }|1/q0≤Ckfk∞, for every t > 0, whih means that the Lorentz norm of b f in the Lorentz spae ℓq0,∞ is dominated by kfk∞ . Now, ℓq0,∞ is ontinuously injeted in ℓq for q > q0 (see for instane [38℄, I I p. 143), and this gives the desired result.  We denote, as usual, by c0 the lassial spae of sequenes x= (xn)n≥0 tending to zero at innity, equipp ed with the norm kxk= supn|xn| . We say, in the usual familiar way, that a Banah spae X ontains c0  if X has a losed subspae isomorphi to c0 . Our notation for Banah spaes is lassial, as an b e found in [16℄, [38℄ or [59℄ for instane. A subset Λ of Z is said to b e a Rosenthal set if every b ounded measurable funtion on T with spetrum in Λ is almost everywhere equal to a ontinuous 8 funtion (in short L∞ Λ=CΛ ). Λ is not Rosenthal if and only if L∞ Λ is not separable, so suh a set an b e thought as being a big set. Every Sidon set is learly Rosenthal, but H.P. Rosenthal gave examples of non-Sidon sets whih are Rosenthal ([53℄). We shall make use of the following well known negative riterion (see [41℄,  3), whih follows from the lassial theorem of C. Bessaga and A. Peªzy«ski ([38℄, I.2.e.8), saying that a dual spae whih ontains c0 has to ontain also ℓ∞ . Prop osition 1.7 If CΛ ontains c0 , then Λ is not a Rosenthal set. Denition 1.8 Let Λ⊆N∗≡N\{0} , and set ΛN= Λ ∩[1, N] and AN(t) = 1 |ΛN|X n∈ΛN en(t). We say that Λ is: - ergo di if (AN(t))N≥1 onverges to a limit lΛ(t)∈C for eah t∈T . - strongly ergo di if it is ergodi and moreover the limit funtion lΛ denes an element of c0(T) : for every ε > 0 the set {t∈T;|lΛ(t)|> ε } is nite. - uniformly distributed if it is (strongly) ergodi and, moreover, lΛ(t) = 0 for t6= 0 mod. 2π . The reason for this terminology is that the ergo di sets are those for whih an ergo di theorem holds: (1/|ΛN|)Pn∈ΛNTn onverges in the strong op erator top ology for every ontration T of a Hilb ert spae. Typially, the set of dth p erfet p owers, or the set of prime numbers are strongly ergo di (aording to the result of Vinogradov for t irrational mo d. 2π , and to the Dirihlet's arithmeti progression theorem for t rational mo d. 2π ). The third name omes from H. Weyl's lassial riterion for the equidistribution of a real sequene mo d. 2π . The relationship b etween these notions omes from: Theorem 1.9 (F. Lust-Piquard [42℄) Let Λ⊆[1,+∞[ be a set of positive integers. Then: (a) If Λ is strongly ergodi, CΛ ontains c0. (b) More general ly, if Λ is strongly ergodi and D⊆Λ has a positive upper density with respet to Λ , then CD ontains c0 as wel l. Here p ositive upp er density means that: lim N→+∞|D∩[1, N]| |Λ∩[1, N]|>0. Part (b) will b e useful to us in the last theorem of Setion 2. See [42℄ for the pro of of this theorem. The underlying idea for (a) is that if AN(t)→lΛ(t) for every t∈T , lΛ denes an element of the biorthogonal C⊥⊥ Λ , and the ondition lΛ∈c0(T) implies that it is the sum of a weakly un- onditionally Cauhy series of ontinuous funtions. By using a p erturbation 9 Put ν=δ0−VMn , where δ0 is the Dira p oint mass at 0 , and VMn the de la Vallée-Poussin kernel of order Mn . Consider the Riesz pro dut R= Q k∈B (1 + Reek) , and set µ= 2ν∗R . We laim that: kµk ≤ 8 ; bµ≥1 on B;bµ= 0 on Λ\B. Indeed, kνk ≤ 4 and B is quasi-indep endent, so the Riesz pro dut R veries kRk=b R(0) = 1 . Therefore kµk ≤ 8 . Take l∈B . Then l > 2Mn and bν(l) = 1 . As b R(l)≥1/2 , we have bµ(l)≥1 . If A⊆Λ and |A|ε≤n1 , any singleton B of A is quasi-indep endent, 1 - omplemented in Λ , and |B| ≥ n−1 1|A|ε . We have thus veried the hyp othesis of part (b) of Prop osition 1.6, and so Λ is q -Sidon for any q > 1/ε . In partiular, it is p -Sidon, and this ends the proof of Theorem 2.2.  Remark 1. The pro of shows that we an atually extrat from A , for every α > 0 , a quasi-indep endent set B suh that |B| ≥ δ|A|/(log |A|)2+α . Moreover, a slight modiation leads to sets even loser to Sidon sets. Prop osition 2.4 Let α > 1 and ϕα be the Orliz funtion x7→ xlog(1+x)α . Then, there exists a set Λ as in Theorem 2.2 , and moreover suh that b f∈ℓϕα,∞ for every f∈ CΛ . Reall that ℓϕα,∞ is the weak Orliz-Lorentz spae of sequenes (an)n suh that sup n ϕ−1 α(n)a∗ n<+∞ , where (a∗ n)n is the non-inreasing rearrangement of (|an|)n . Therefore, another way to phrase the prop osition is, setting an=b f(n) : a∗ n≤Cαkfk∞(log n)α/n for every f∈ CΛ. The pro of just onsists in hanging Mn . We take Mn=en(log log n)2 , where [ ] stands for the integer part. We still have P n PΩn(Mn)<+∞ , sine PΩn(Mn)≤2Cn nn (log Mnlog log Mn)n Mn≤exp −n(log log n)2/2 for n large enough. Arguing as previously, we get for every nite subset A of Λ , a quasi-indep endent subset B of A suh that |B| ≥ δ|A|/(log |A|)α , and suh that CB is uniformly pseudo-omplemented in CA . As in the pro of of Proposition 1.6, we obtain |{|b f|> t}| ≤ C ϕαkfk∞ t, whih gives the result (arguing as in [34℄ for instane). We annot eliminate a logarithmi fator, and replae α > 1 by α > 0 b eause, due to Bourgain's riterion, we have to assume that σn/log n go es to 16 innity in order that CΛ ontains c0 . However, for eah α > 0 , there do exist non-Sidon sets Λ for whih b f∈ℓϕα when f∈ CΛ (as an b e seen from [5℄, p. 69). The set Λ is, in some sense, very lose to b e Sidon, whereas CΛ ontains c0 . However, it annot be to o lose without b eing Sidon beause if b f∈ℓ1,∞ , the Lorentz spae weakℓ1 , for every f∈ CΛ , then Λ is Sidon. In fat, this ondition implies an inequality of the type: |{|b f| ≥ t}| ≤ C tkfk∞(∗) for every f∈ CΛ . Let now A b e a nite subset of Λ , and f=P n∈A en and fω=P n∈A rn(ω)en , where rn , n≥1 are the Rademaher funtions. Then, inequality ( (∗) ) applied with t= 1 gives kfωk∞≥(1/C)|A| . Integrating in ω gives [[f]] ≥(1/C)|A| , from whih follows, by a result of G. Pisier ([48℄, Théorème 2.3 (vi)), that Λ is a Sidon set. Remark 2. If one takes seletors of mean δn suh that nδn is b ounded, the orresp onding random set Λ(ω) is almost surely a Sidon set. This is a well-known result of Y. Katznelson and P. Malliavin ([28℄, or [27℄), and Lemma 2.1 gives another pro of of this fat. It sues to take Mn=An , where A is a given integer, large enough to have P+∞ n=1 PΩn(Mn)<1 . Then, with positive probability Λ(ω)∩[Mn,+∞[ ontains no relation of length ≤n , whereas |Λ(ω)∩[1, Mn]| ≤ Cn . Hene, for every nite subset A of Λ(ω) , we an nd a quasi-indep endent subset B⊆A suh that |B| ≥ δ|A| , for some xed δ=δ(ω) . It follows from Pisier's haraterization ([48℄, Th. 2.3 (iv)) that, with p ositive probability, and hene almost surely by Kolmogorov's 0−1 law, Λ(ω) is a Sidon set. As is now well-known, Sidon sets are haraterized by various prop erties (suessively weaker) of the Banah spae CΛ : Λ is a Sidon set i CΛ is isomorphi to ℓ1 ([58℄), i CΛ has otyp e 2 ([31℄, Th. 3.1, [46℄), and i CΛ has a nite otyp e ([13℄). This later prop erty an b e expressed by saying that CΛ do es not ontain ℓn ∞ uniformly. So, deterministially, one has the dihotomy: (a) either Λ is a Sidon set, and so CΛ is isomorphi to ℓ1 ; (b) or CΛ ontains ℓn ∞ uniformly. The probabilisti dihotomy is stronger: taking seletors of mean δ1, δ2,... , with (δn)n dereasing, one has: (a) either almost surely Λ is a Sidon set (if nδn is b ounded); (b) or almost surely CΛ ontains c0 (if nδn is not bounded), and Λ is even uniformly distributed. Y. Katznelson ([27℄) already notied suh a dihotomy: he showed that (under a dierent hoie of seletors from ours) either almost surely Λ is a Sidon set, or almost surely Λ is dense in the Bohr group. However, this is p erhaps not a true dihotomy sine it is a well-known op en problem whether there an exist Sidon sets dense in the Bohr group (see [15℄, question 2, p. 14; 17 it is stated for the Bohr group of R , but also makes sense for the Bohr group of Z ). The dihotomy stated here strengthens Katznelson's result sine every uniformly distributed set is dense in the Bohr group (see [6℄, Theorem 1); indeed, saying that Λ = {λ1, λ2,...} is uniformly distributed means that the measures µN= 1/N +∞ P n=1 δλn ( δλn is there the Dira measure at the point λn ) onverge weak-star to the Haar measure µ of the Bohr group bZ ; but these measures are arried by Λ , so the losed supp ort of µ is ontained in the Bohr losure of Λ ; sine the Haar measure is ontinuous, we get that this losure is the whole Bohr group. Remark 3. The random sets Λ that we onstrut have an asymptotial quasiindep endene: Λ∩[Mn,+∞[ ontains no relation of length ≤n . This is reminis- ent of the following result of J. Bourgain ([7℄): if Λ is a Sidon set and n∈N∗ , there exists ln=l(Λ, n) suh that Λ an b e deomp osed in ln sets Λ1,...,Λln , eah of whih ontains no relation of length ≤n . We now investigate what happ ens when we let p inrease away from 1 .We get several dierent results, and p= 4/3 seems to play a sp eial role. We rst state two very similar results. Theorem 2.5 For every 1< p < 4/3 , there exists a set Λ of integers whih is: (1) uniformly distributed (so Λ is dense in the Bohr group, CΛ ontains c0 , and Λ is not a Rosenthal set), and whih is: (2) Λ(q) for al l q < +∞ , a CUC -set, and moreover is: (a) p -Rider, but not q -Rider for q < p (b) q -Sidon for al l q > p/(2 −p) . Theorem 2.6 Same as Theorem 2.5, exept that, instead of (a) , Λ is: (a') q -Rider for every q > p , but is not p -Rider. Remark. After this pap er was ompleted, P. Lefèvre and the third-named author proved ([36℄) that every p -Rider set with p < 4/3 is a q -Sidon set, for all q > p/(2 −p) . A weaker, unpublished, result, due to J. Bourgain, is quoted in [15℄, p. 41. Hene ondition (b) always follows from ondition (a), and is not sp ei to the onstrution. We do not know whether this gap b etween p and p/(2 −p) follows only from tehnial reasons. For p > 1 , whether every p -Rider set is atually p -Sidon is an op en question. In Theorem 2.5, we obtain sets whih are p -Rider but not q -Rider for q < p . We do not know if these sets are p -Sidon, so exatly p -Sidon, in the terminology of R. Blei. He onstruted suh sets using frational pro duts ([3℄, [4℄). We may all the sets in Theorem 2.5  exatly p -Rider sets . The sets app earing in Theorem 2.6 are of a dierent kind. We may all them  exatly p+ -Rider sets . Suh sets were also obtained in [3℄, Corol. 1.7 d), where they were alled exatly nonp -Sidon, and were alled asymptoti p -Sidon in [5℄. 18 Pro of. It is similar to that of Theorem 2.2, so we shall be very skethy. Let α= 2(p−1)/(2 −p)∈] 0,1 [ . For Theorem 2.5, we use seletors εk of mean δk=c(log k)α k(log log k)α+1 for k≥4. As in Lemma 2.3, we have, with Mn=nn , Pn≥1PΩn(Mn)<+∞ , and almost surely C0nα+1 ≤ |ΛMn| ≤ C1nα+1 and |Λ′ n| ≤ C1nα for n large enough. For Theorem 2.6, we inrease the means δk slightly, replaing them by δk=c(log k)αlog log k k· Remark. In order to prove our theorems, we used seletors with various means. They are smaller in Theorem 2.2 than in Theorem 2.5, for instane. We remark that seletors (εk)k of mean δk with δk≤δ′ k may b e ahieved as the pro dut of two indep endent sequenes of seletors (ε′ k)k and (ε′′ k)k of mean δ′ k and δ′′ k= δk/δ′ k . It follows that, for example, the sets in Theorem 2.2 may b e onstruted inside the resp etive sets of Theorem 2.5. In Theorem 2.5, the pro of that Λ was CUC or Λ(q) was based on the fat that |Λ′ n| ⊆ Λ∩[Mn,+∞[ is quasi-indep endent. For α≥1 ( i.e. p≥4/3 ), we no longer have |Λ′ n| ≤ n , and therefore, a priori , must give up these prop erties. However, we an use another extration proedure. This proedure was rst intro dued by J. Bourgain ([8℄); later, a lear statement was given in [52℄,  I I I.2. Sine this last referene is hardly available, we prefer to give a self-ontained pro of. The orresp onding set Λ(ω) of integers that we shall obtain in this manner satises |Λ(ω)∩[2n,2n+1[| ∼ n∼log 2n , whih is the limiting ondition of mesh (on arithmeti progressions) for Sidon sets. This size is in some sense the largest p ossible if we want to obtain a set Λ with blo ks having a uniformly b ounded Sidon onstant. Theorem 2.7 There exists a set Λ of integers whih is uniformly distributed and ontains a subset E⊆N∗ whih is: (1) 4/3 -Rider, and not q -Rider for q < 4/3 ; a CUC -set; a Λ(q) -set for al l q < +∞ (more preisely, for al l q > 2 , we have: kfkq≤Cq2kfk2 for al l f∈ PE , where C > 0 is a numerial onstant), and nevertheless, (2) has positive upper density in Λ , so, in partiular, CE ontains c0 , and E is not a Rosenthal set. Let A b e a nite subset of integers. For the pro of, it will be onvenient to dene: ψA= sup p≥2 keAkp √p, where eA=X k∈A ek. We need the following simple estimate of ψA . 19 Lemma 2.8 Let I= [a+1, a+N] be an interval of integers of length N , N≥3 . Then: ψI≤N √2 log N· Pro of. For p≥2 , |eI|p≤Np−2|eI|2 , so R|eI|pdm ≤Np−2R|eI|2dm =Np−1 and keIkp/√p≤N1−1/p/√p . Optimizing gives p= 2 log N ( ≥2 ), and the lemma.  This estimate is essentially optimal. Indeed, it is well-known that ψI is uniformly equivalent to θ=keIkΨ ( k kΨ b eing the norm asso iated to the Orliz funtion Ψ(x) = ex2−1 ). But, for some onstant γ , |eI(t)| ≥ γN for t in an interval J of length ≥γN−1 around 0 , so one has: 2≥ZJ exp |eI|2 θ2dm ≥γN−1exp γ2N2 θ2, whene θ≥γ−1N/plog 2γ−1N . We now use seletors εk of mean δk=c n/2n for 2n≤k < 2n+1 , where c > 0 is a given onstant. Set In= [2n,2n+1[, n ≥2 ; δk=cn 2n if k∈In. Note that (δk)k dereases, and δk is of the form αk/k , where (αk)k go es to +∞ . If Λ = Λ(ω) is the orresp onding set of integers, it will b e onvenient to set: Λn= Λ ∩In;σn=E|Λn|=X k∈In δk=cn . For this pro of, the value of ψIn is somewhat large, and requires c b e su- iently small, say c≤1/576 . We prefer to follow another route, whih ould b e useful in other ontexts, by ho osing also a random set in In for whih the ψ onstant is small enough. We make the two random hoies at the same time. Namely, we onsider (ε′ n)n≥1 , a seond sequene of seletors, indep endent of (εn)n≥1 , with xed mean τ , and set Λ′ n(ω) = {k∈Λn(ω) ; ε′ k(ω) = 1} . In short: Λ′ n={k∈Λn;ε′ k= 1}; Λ′= +∞ [ n=1 Λ′ n. The following lemma, whih is a slight modiation of Bourgain's onstru- tion in [8℄, is really the heart of the proof. Lemma 2.9 Almost surely, for n large enough, one has: (1) (c/2) n≤ |Λn| ≤ (2c)n and (cτ/2) n≤ |Λ′ n| ≤ (2cτ)n (2) Λ′ n ontains at most relations of length ≤ln , where ln= [144 c2τ2n] . 20 Pro of of Lemma 2.9. We have already seen that: P|Λn|−σn≥σn 2≤exp −σn 32 = exp −c n 32 , so, by the Borel-Cantelli lemma, |Λn| is almost surely b etween (c/2) n and (2c)n for n large enough; and this proves the rst half of (1). The seond half holds for the same reason, sine Λ′ n orresp onds to seletors εkε′ k with mean (cτ)n/2n for k∈In . The pro of of (2) is more elab orate. Fix n , and onsider the random trigonometri p olynomial: Fω= |In| X j=ln+1 X R⊆In |R|=jY k∈R εk(ω)ε′ k(ω)ek+e−k. Set: Nn(ω) = ZT Fω(t)dm(t). Expanding Fω , we see that: Fω(t) = |In| X j=ln+1 X R⊆In |R|=j X θk∈{−1,+1}RY k∈R εk(ω)ε′ k(ω)eθk k(t) = |In| X j=ln+1 X R⊆In |R|=jX θk∈{−1,+1}R eitPk∈Rθkk. The ontribution to Nn(ω) of an exp onential of this sum is 0 if P k∈R θkk6= 0 , and is 1 if P k∈R θkk= 0 . Therefore, Nn(ω) is exatly the number of relations of length > ln in Λ′ n . We laim that Nn(ω) is almost surely zero for n large enough. To that eet, we ma jorize the exp etation J of Nn(ω) , using Fubini's theorem. Indeed, J=RTH(t)dm(t) , where: H(t) = ZΩ Fω(t)dP(ω) = |In| X j=ln+1 X R⊆In |R|=j δjY k∈R (ek+e−k). and δ=cτ n/2n . Hene: J= |In| X j=ln+1 X R⊆In |R|=j δjZTY k∈R(ek(t) + e−k(t)dm(t). 21 At this stage, it is useful to observe that: X R⊆In |R|=j ZTY k∈Rek(t) + e−k(t)dm(t) ≤1 j!ZTX k∈In(ek(t) + e−k(t)jdm(t). (3) Indeed, when we expand X k∈In(ek(t) + e−k(t)j, eah term Qk∈Rek(t) + e−k(t) app ears j! times, whereas the other terms on the right hand side of (3) are p ositive. It now follows from (3) that: J≤ |In| X j=ln+1 δj j!ZTX k∈In(ek(t) + e−k(t)jdm(t) ≤ |In| X j=ln+1 δj j!2j  X k∈In ek   j j≤ |In| X j=ln+1 2jδj j!(ψInpj)j. Sine j!≥(j/e)j≥(j/3)j , this gives J≤ +∞ X j=ln+1 6δψIn √jj≤ +∞ X j=ln+1 6δψIn √ln+ 1j. Therefore, J≤2−ln if 6δψIn √ln+ 1 ≤1 2 , i.e. if ln+ 1 ≥144(δψIn)2 . But, it follows from Lemma 2.8 that: ψIn≤2n p(2 log 2) n≤2n √n· Therefore 144(δψIn)2≤144cτ n 2n·2n √n2= 144 c2τ2n, and the hoie of ln just ts to obtain J≤2−ln . Of ourse, we have assumed n large enough to have ln≥1 in that pro of. Finally, Markov's inequality implies: X n≥2 P(Nn≥1) ≤X n≥2 ENn≤X n≥2 2−ln<+∞, 22 and by the Borel-Cantelli lemma, the integer Nn is almost surely zero for n large enough, and that ends the pro of of Lemma 2.9.  Now, using Bourgain's Theorem 1.10 and Lemma 2.9, one an nd Ω0⊆Ω with P(Ω0) = 1 suh that for ω∈Ω0 , there exists n0=n0(ω) suh that Λ = Λ(ω) and Λ′= Λ′(ω) satisfy: (4) Λ and Λ′ are uniformly distributed (5) (c/2) n≤ |Λn| ≤ (2c)n and (cτ/2) n≤ |Λ′ n| ≤ (2cτ)n for n > n0 (6) Λ′ n ontains at most relations of length less than ≤ln= [144c2τ2n] for n > n0 . Λ′ n is not quite quasi-indep endent, so we shall mo dify it slightly. We adjust one and for all τ , dep ending on c , suh that 144c2τ2≤cτ/4 ( e.g. taking cτ = 1/576 ), so that ln≤cτ n/4≤ |Λ′ n|/2 for n > n0 , in view of (5). Selet then in Λ′ n a relation R with supp ort Sn of maximal ardinality. Then |Sn| ≤ ln from (6), and En= Λ′ n\Sn is quasi-indep endent. Moreover: |En|=|Λ′ n|−|Sn| ≥ |Λ′ n|−ln≥ |Λ′ n|/2 for n > n0 . Hene, if we set E=S n>n0 En , we have En=E∩In , and, moreover: (7) E has p ositive upp er density in Λ (note that Λ′ has upp er density ≥τ/4 in Λ by (5)), (8) En is quasi-indep endent, (9) |En| ≥ (cτ/4) n , (10) If A⊆E is a nite subset, then A ontains a quasi-indep endent subset B with |B| ≥ (1/2) |A|1/2 . The last prop erty is proved in the following way. Set Z={n;A∩En6=∅} and h=|Z| . We distinguish two ases. Case 1: there exists n∈Z suh that |A∩En| ≥ |A|1/2 . Then, just take B=A∩En to have a quasi-indep endent set B suh that |B| ≥ |A|1/2 . Case 2: |A∩En|<|A|1/2 for any n∈Z . Then h≥ |A|1/2 . Write Z={n1<···< nh} , and pik an integer mj∈ A∩Enj for eah j= 1,...,h . Then B={m1, m3,...}=: {µ1, µ2,...} is quasiindep endent b eause we have µj+1/µj≥2 . Moreover |B| ≥ h/2≥(1/2) |A|1/2 . It is now easy to see that E has the required properties. Indeed, it follows from (4), (7), and from F. Lust-Piquard's Theorem 1.9 that E has a p ositive upp er density in Λ . That it is CUC follows from (8) and from Proposition 1.2. That it is Λ(q) for all q < +∞ follows from (8) and from Prop osition 1.1. 23 The fat that E is 4/3 -Rider follows from (a) in Prop osition 1.6. Indeed, if ε(p) = 2/p −1 , then ε(4/3) = 1/2 . Finally, let N b e a large integer, and n suh that 2n≤N < 2n+1 . Then |E∩[1, N]| ≥ |En0+1|+···+|En−1| ≥ cτ 4(n0+ 1) + ···+ (n−1) ≥d0n2≥d1(log N)2 where d0, d1 are p ositive onstants. If now E is a p -Rider set, we have the mesh ondition |E∩[1, N]|=O(log N)p/(2−p) . This requires 2≤p/(2 −p) , that is p≥4/3 . And this ends the proof of Theorem 2.7.  Remark. The third-named author proved the following ([52℄, Lema 2.4) (whih is atually impliitly already ontained in [48℄, Lemme 7.2, Théorème 7.1, and Théorème 2.3 (iv)): ( ∗ ) For every nite subset A⊆Z , there exists a quasi-indep endent subset B⊆A suh that |B| ≥ δ(|A|/ψA)2 , where δ > 0 is a numerial onstant. On the other hand, G. Pisier ([47℄, Lemme 5.2) proved: E  X k akrkek Ψ≤CX k|ak|21/2 (1) where C is a numerial onstant, (rk)k is the Rademaher sequene, and k kΨ is the Orliz spae asso iated to Ψ(x) = ex2−1 . Taking our seletors εk with mean δk=c n/2n for k∈In , standard symmetrization and entering arguments give: E  X k∈In εkek  Ψ≤C√n . (2) In other terms, we have, in view of Lemma 2.9: E(ψΛn)≤C√n≤C′|Λn|1/2. (3) If we ould prove a onentration inequality, variant of Lemma 1.3, then this variant and the Borel-Cantelli lemma would imply from (3) that: Almost surely ψΛn≤C′′|Λn|1/2 for n large enough. (4) We ould then ombine ( ∗ ) and (4) diretly to obtain the following alternative pro of of Theorem 2.7. Selet ω∈Ω suh that Λ is strongly ergo di, with |Λn| ≥ c n/2 , and ψΛn≤C′′|Λn|1/2 ; take then a quasi-indep endent set En⊆Λn of size |En| ≥ δ|Λn| ψΛn2 ≥δC′′−2|Λn| ≥ δ′n; the set E=S n En then has the required prop erties. 24 To end this setion, we onsider the ase p > 4/3 . We annot keep the prop erty of uniform onvergene ( CUC ), nor that of b eing q -Sidon stated in Theorem 2.5. We do not know whether this is only due to the metho d. But b eing p -Rider with p > 4/3 might b e a rather weak ondition (see [35℄ and [36℄). Theorem 2.10 For every 4/3≤p < 2 there exists a set Λ of integers whih is p -Rider, but is not q -Rider for q < p and whih is Λ(q) for every q < +∞ , but whih is uniformly distributed (so in partiular dense in the Bohr group, and CΛ ontains c0 ). The pro of is essentially the same as in Theorem 2.2, exept that we take seletors εk of mean δk=c(log k)α k(log log k)α+1 for k≥1, where α= 2(p−1)/(2 −p)≥1 , and replae Mn=nn by the smallest integer ≥nβn , with β any numb er > α (for instane β=α+ 1 ), whih we all again Mn . The estimate: PΩn(Mn)≤2cn nn (log Mn)n(α+1) Mn still holds, and now gives: PΩn(Mn)≤C′n(log n)n(α+1) n(β−α)n· Then easy omputations show that: ( ∗ ) Almost surely |ΛMn| ∼ (nlog n)α+1 for n suiently large; ( ∗∗ ) Almost surely |Λ′ n| ∼ nα(log n)α+1 for n suiently large. Prop erty ( ∗ ) guaranties that Λ(ω) will still b e almost surely p -Rider, and ( ∗∗ ) with the mesh ondition implies that Λ is not q -Rider for q < p . 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