Some Diophantine properties of the sequence of S-units

SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS ATTILA B´ ERCZES, ANDREJ DUJELLA, AND LAJOS HAJDU Abstract. We prove some diophantine properties of the sequence of S-units. 1. Introduction Integers having no prime factors outside a fixed set of primes play important role and are heavily investigated in several parts of number theory. For example, they play special role in diophantine number theory; see e.g. the classical survey paper of Evertse, Gy˝ory, Stewart and Tijdeman [1] or Chapter 1 of the book of Shorey and Tijdeman [7] and the references given there. Further, the sequence formed of such integers is also of interest. To be precise, fix primes p1<· · · <pt, and write snfor the sequence of integers composed of these primes, arranged in an increasing order. Tijdeman [8] and [9] provided sharp upper and lower bounds for the gaps between consecutive terms of the sequence, respectively. These bounds have the nice property that they are ”almost” equal. Namely, Tijdeman proved that (1.1) sn (log sn)c1< sn+1 −sn<sn (log sn)c2 hold with some effectively computable absolute constants c1and c2for all index nwhich is large enough. In the proofs of both the lower and the upper 2010 Mathematics Subject Classification: Primary 11B83; Secondary: 11N25, 11J70. Keywords and Phrases: S-unit, integers divisible by fixed primes, continued fraction. The research was supported in part by grants K67580, K75566, K100339 (A.B., L.H.) and NK101680 (L.H.) of the Hungarian National Foundation for Scientific Research. The work is supported by the T´ AMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund. (A.B., L.H.). 1 2 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU bound in (1.1) the approximation properties of the tuple (log p1, . . . , log pt) play a crucial role. These are mainly used through Baker’s theory, but in establishing the upper bound also the continued fractions of log pi/log pj play a vital role. In this paper we develop a method to explicitly give the gaps in the sequence sn. In other words, for any term snwe can find both sn−1and sn+1, at least in principle, without enumerating all terms of the sequence. Again, here the approximation properties of the tuple (log p1, . . . , log pt) are decisive. In the case when there are two fixed primes, we even give an efficient and algorithm to find these terms explicitly. This is done by the careful analysis of the behavior of the continued fractions of log p1/log p2. Since to explain our results and methods in detail we need several notions and notation, we shall do that in the next section. 2. Main results Let S={p1,p2, . . . , pt}be a set of trational primes, and in the sequel suppose that p1<p2<· · · <pt. The ring of rational S-integers is denoted by ZS, and its unit group by Z∗ S. Consider those S-units, which are natural numbers, and denote by (sn) the sequence consisting of these numbers in increasing order. Clearly, any element of the sequence (sn) can be written in the form sn=pcn,1 1pcn,2 2. . . pcn,t twith cn,i ∈Z≥0. Consider the hyperplane P ⊂ Rtdefined by P:= {(x1, . . . , xt) : x1log p1+· · · +xtlog pt= 0}. Then Pis a subspace of Rt, in particular, it clearly contains the origin. For a point a= (a1, . . . , at)∈Zt ≥0denote by d(a) the Euclidean distance of the point afrom the hyperplane Pin Rt. Theorem 2.1. The following statements are true: (i) For all a= (a1, . . . , at),b= (b1, . . . , bt)∈Zt ≥0we have pa1 1. . . pat t<pb1 1. . . pbt t⇐⇒ d(a)< d(b). In particular, d(a) = d(b)if and only if a=b. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 3 (ii) Let r∈R>0, and write c(r) := log r √log2p1+· · · + log2pt . Then the smallest snfor which sn> r is that sn=pa1 1. . . pat twith a= (a1, . . . , at)∈Zt ≥0for which for every b= (b1, . . . , bt)∈Zt ≥0 with d(b)> c(r)we have c(r)< d(a)< d(b). Similarly, the largest snfor which sn< r is that sn=pa1 1. . . pat twith a= (a1, . . . , at)∈Zt ≥0for which for every b= (b1, . . . , bt)∈Zt ≥0 with d(b)< c(r)we have c(r)> d(a)> d(b). Further, in both cases acan be effectively determined. Remark. The proof of Theorem 2.1 is based upon some properties of a certain special multidimensional diophantine approximation. For the theory of multidimensional diophantine approximations of different types see the excellent survey paper of Moshcevitin [4], and the references given there. In the special case t= 2 we can formulate much more precise results. In order to do so, we need to introduce some further notation. From now on let S={p,q}be a set of two rational primes with p<q. Now the sequence (sn) may be written in the form sn=pcnqdnwith cn, dn∈Z≥0. We define the companion sequence (fn) of (sn) by (2.1) fn:= dn+1 −dn cn−cn+1 . Later we shall prove that the elements of the sequence (fn) are always well defined (i.e. cn−cn+1 = 0), they are always in lowest terms (i.e. gcd(dn+1 −dn, cn−cn+1) = 1), and fn≥0, with equality precisely for values of nfor which sn<q. In the statement of our results below we use notions related to the continued fractions of real numbers. Here we use these notions without any reference, however the concepts and results connected to continued fractions which are needed in the paper, are summarized in Section 3. 4 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Given a concrete element of the sequence (sn), the following theorem gives a simple algorithm how to determine the next element in the sequence. Theorem 2.2. Let the sequences (sn),(cn),(dn)and (fn)have the same meaning as above. Suppose that we are given sk=pckqdk. Then we can compute sk+1 in the following way: •Let u1 v1be the upper convergent of log p log qwith maximal denominator for which v1≤ckholds. •Let u2 v2be the lower convergent of log p log qwith maximal numerator for which u2≤dkholds. •Put x:= |v1log p−u1log q| − |v2log p−u2log q|and (2.2) ck+1 =   ck−v1if x < 0, ck+v2if x > 0,dk+1 =   dk+u1if x < 0, dk−u2if x > 0. Then we have sk+1 =pck+1 qdk+1 . Remark. In view of the method of the proof, having skone can explicitly give the term sk−1of the sequence, similarly to the term sk+1. However, since in the light of Theorem 2.2 this can be done in the obvious way, we omit the details. In the following theorem we summarize basic properties of the companion sequence, which sequence describes how the exponents of pand qchange when we move from snto sn+1. Theorem 2.3. Let the sequences (sn),(cn),(dn)and (fn)have the same meaning as above. Then we have the following properties: (i) The sequence (fn)is well-defined, i.e. cn+1 =cnfor all n∈N. (ii) We have fn≥0for all n∈N, with equality precisely for those values of nfor which sn<q. (iii) All companion fractions fnare convergents of log p log q, and •if fnis an upper convergent then ck+1 < ckand dk+1 > dk, •if fnis a lower convergent then ck+1 > ckand dk+1 < dk. (iv) Suppose that the smallest index nsuch that fn=u vis k. Then •if u vis an upper convergent of log p log qthen we have sk=pvand sk+1 =qu; SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 5 •if u vis a lower convergent of log p log qthen we have sk=quand sk+1 =pv. Conversely, •if sk=pvand sk+1 =quthen fk=u vis an upper convergent of log p log qand kis the index of u vin the sequence (fn); •if sk=quand sk+1 =pvthen fk=u vis a lower convergent of log p log qand kis the index of u vin the sequence (fn). (v) Let pi,j qi,j be a convergent of log p log q. The number of occurrences of pi,j qi,j in the sequence (fn)is exactly pi+1qi+1, where pi+1 qi+1 is the principal convergent of log p log qfollowing the principal convergent pi qi=pi,0 qi,0. To understand well the structure of our sequence (sn) we need to know how the corresponding companion sequence (fn) behaves. Some of the most important arising questions are the following: •if we know the value of fnthen which values can be taken by fn−1 and fn+1 respectively •how many consecutive elements of the sequence fnmay have the same value pi,j qi,j . Theorems 2.4 and 2.5 give a precise answer to these questions. In one hand we prove that an intermediate convergent cannot be the value of two consecutive elements of (fn), and that there are at most aj+2 + 1 consecutive elements of (fn) which assume the same value pj qj. Further our Theorems describe all possible patterns formed by exactly k(1 ≤k≤aj+2 + 1) consecutive elements of (fn) assuming the same value pj qj, and by the preceding and the following elements. Moreover, our Lemmas in Section 6 give necessary and sufficient conditions for cn= ordqsnand dn= ordpsnso that sn−1 is the starting point of such a concrete pattern. In the following Theorem 2.4 we answer the above question for principal convergents, and in Theorem 2.5 we do the same for intermediate convergents. Theorem 2.4. Let us suppose that in the sequence of companion fractions we have the following pattern: (2.3) fn−1=pl ql , fn=fn+1 =· · · =fn+k−1=pl ql , fn+k=pl ql . 6 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Then we have 1≤k≤al+1 + 1, and for (fn−1, fn+k)we have the following possibilities: (i) If 1≤k < al+1 then (2.4) (fn−1,fn+k)∈{(pl+1 ql+1 ,pl+1 ql+1 ),(pl−1,k−1 ql−1,k−1 ,pl−1,k−1 ql−1,k−1), (pl−1,k−1 ql−1,k−1 ,pl−1,k ql−1,k )(pl−1,k ql−1,k ,pl−1,k−1 ql−1,k−1)(pl−1,k ql−1,k ,pl−1,k ql−1,k )} (ii) If k=al+1 then (2.5) (fn−1, fn+k)∈{(pl+1 ql+1 ,pl+1 ql+1 ),(pl−1,k−1 ql−1,k−1 ,pl−1,k−1 ql−1,k−1), (pl−1,k−1 ql−1,k−1 ,pl+1 ql+1 )(pl+1 ql+1 ,pl−1,k−1 ql−1,k−1)} (iii) If k=al+1 + 1 then (2.6) (fn−1, fn+k) = (pl+1 ql+1 ,pl+1 ql+1 ). Theorem 2.5. Suppose that fn=pl,j ql,j with some 1≤j < al+2 (i.e. fnis an intermediate convergent). Then we have (2.7) fn−1=fn+1 =pl+1 ql+1 . 3. Continued fractions In this section we summarize important properties of the continued fraction expansion and the corresponding convergents of real numbers. For the general theory of continued fractions we refer to the classical books [2], [5], [6] and the references given there. If S={p,q}, then, as we have seen, the structure of the sequence of natural S-units is strongly connected to the convergents of the real number log p log q. The proofs of the properties listed below may be found in [2], [5] and [6]. Let 0 =α∈Rbe a real number and define a0, a1, a2, . . . in the following way: α0:= α,a0:= [α0], αi+1 := {1 {αi}},ai+1 := [ 1 αi+1 ], . The sequence (an) is called the continued fraction of α. In the sequel, for 0 =α∈Rwe shall denote by [a0, a1, a2...] the continued fraction expansion of α. Put (3.1) p−2= 0, p−1= 1, pi=aipi−1+pi−2(i≥0) SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 7 and (3.2) q−2= 1, q−1= 0, qi=aiqi−1+qi−2(i≥0). The fractions pi/qifor i≥0 are called the principal convergents of α. Further, for non-negative integers iand jput (3.3) pi,j =jpi+1 +pi, qi,j =jqi+1 +qi. The fractions (3.4) pi,j qi,j =jpi+1 +pi jqi+1 +qi 1≤j≤ai+2 −1 are called the intermediate convergents of α. We mention, that in many cases it is comfortable to let in (3.4) the index jassume also the values 0 and ai+2, in these cases the resulting fraction in (3.4) being a principal convergent, namely: (3.5) pi,0 qi,0 =pi qi and pi,ai+2 qi,ai+2 =pi+2 qi+2 . The principal convergents and intermediate convergents together are called convergents. For the convergents of αwe have the following properties: · · · <pi qi <···<pi,j qi,j <pi,j+1 qi,j+1 <· · · <pi+2 qi+2 < . . . if iis even,(3.6) · · · >pi qi >···>pi,j qi,j >pi,j+1 qi,j+1 >· · · >pi+2 qi+2 >· · · >if iis odd,(3.7) and pi,j−1qi,j −pi,jqi,j−1= (−1)jfor i≥0 and 1 ≤j≤ai+2 −1. In the sequel the fractions (3.6) of even indices will also be referred to as lower convergents, while the fractions (3.7) of odd indices as upper convergents. This terminology is clearly justified by the fact, that lower convergents of αare smaller then α, while upper convergents of αare larger then α. We say that •the rational number p qis a best approximation to αif for every rational number b cwith denominator c < q we have (3.8) |qα −p|<|cα −b| 8 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU •the rational number p qis a best lower approximation to αif p q< α and for every rational number b c< α with denominator c<qwe have (3.9) qα −p < cα −b •the rational number p qis a best upper approximation to αif p q> α and for every rational number b c> α with denominator c<qwe have (3.10) p−qα < b −cα. The first statement of the following lemma is a well-known property of principal convergents (see e.g. [2], [5], [6]), while the second and third statements are due to Kimberling [3]. Lemma 3.1. Let α= 0 be a real number, and denote by pi/qifor i≥0 the principal convergents of αand by pi,j qi,j for i≥0,1≤j < ai+2 the intermediate convergents of α. Then the following statements are true: (i) If b csatisfies |cα −b|<|qiα−pi|then c≥qi+1 (ii) The best lower approximates to αare the lower convergents to α, i.e. the fractions pi,j qi,j for even iand 0≤j < ai+2. (iii) The best upper approximates to αare the upper convergents to α, i.e. the fractions pi,j qi,j for odd iand 0≤j < ai+2. Remark. The first statement of Lemma 3.1 implies as a simple corollary that the best approximates to αare the principal convergents of α. In the last two statements of Lemma 3.1 among the best lower and upper approximations pi,j qi,j for 0 ≤j < ai+2 we can find the principal convergents, i.e. the fractions with j= 0, and the intermediate convergents, i.e. the fractions with 1 ≤j < ai+2. The next lemma is a classical result for continued fractions again (see e.g. [2], [5], [6]). Lemma 3.2. Suppose that p q= 0 is a convergent to a positive real number α. Then q pis a convergent to 1 α. The parity of the index of q pamong the convergents of 1 αis opposite to the parity of the index of p qamong the convergents of α. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 9 4. Applications In this section we give some diophantine applications of our results. Theorem 4.1. There exist infinitely many indices ksuch that the terms sk, sk+1, sk+2, sk+3 form a geometric progression. Proof of Theorem 4.1. In fact we prove more. First note that since α:= log p log q is transcendental, the continued fraction expansion of αcontains infinitely many terms >1, so there are either infinitely many odd values of nwith an+1 >1 or there are either infinitely many even values of nwith an+1 >1. First suppose that there are infinitely many odd values of nwith an+1 >1 and take a fixed odd index nsuch that an+1 >1. Observe that then we have qn+1 =an+1qn+qn−1≥2qn+qn−1, and pn+1 =an+1pn+pn−1≥2pn+pn−1. Choose integers Aand Bsubject to the following restrictions: (4.1) 3qn≤A < 3qn+qn−1,0≤B < pn−1. We claim that with any of the above choices for Aand B, writing sk= pAqBwe have fk=pn qn, and the terms sk, sk+1, sk+2, sk+3 form a geometric progression. To check these assertions, observe that both A < qn+qn+1 ≤min{qn+2, qn,1}and B+ 2pn< pn+1 holds. Hence by Theorem 2.2 we clearly get that sk=pAqB, sk+1 =pA−qnqB+pn, sk+2 =pA−2qnqB+2pn, sk+3 =pA−3qnqB+3pn is a desired geometric progression. Since by our assumption there are infinitely many indices nhaving the desired property, the statement follows. Now we also have to deal with the case when there are only finitely many odd values of nwith an+1 >1. However, in this case there are infinitely many even values of nwith an+1 >1 and choosing any such na similar construction is possible as above, just we have to choose Aand Bsubject to the restrictions 0≤A < qn−1,3pn≤B < 3pn+pn−1, 16 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU 6. Proof of Theorems 2.4 and 2.5 In order to prove Theorem 2.4 and 2.5 we need to separate the cases where lis odd and lis even. Here we only prove the case when lis odd and we mention that the other case can be proved in the very same way. During the proofs we shall use (3.3) several times without further reference. For the rest of this section put l:= 2i+ 1. First we prove Theorem 2.5, since its proof is much simpler. Proof of Theorem 2.5. Lemma 5.1 shows that if 1 ≤j < a2i+3 then fn= p2i+1,j q2i+1,j is equivalent to (6.8) {0≤dn< p2i+2 q2i+1,j ≤cn< q2i+1,j+1. Further, fn=p2i+1,j q2i+1,j also yields cn+1 =cn−q2i+1,j and dn+1 =dn+p2i+1,j. These, together with (6.8) show that we have (6.9) {p2i+1,j ≤dn+1 < p2i+2 +p2i+1,j 0≤cn+1 < q2i+1,j+1 −q2i+1,j, this latter being equivalent to (6.10) {jp2i+2 +p2i+1 ≤dn+1 <(j+ 1)p2i+2 +p2i+1 0≤cn+1 < q2i+2. Now using 1 ≤j < a2i+3 (6.10) has the consequence (6.11) {p2i+2 ≤dn+1 < p2i+3 +p2i+2 0≤cn+1 < q2i+3, which proves (6.12) fn+1 =p2i+2 q2i+2 . Now we prove the statement fn−1=p2i+2 q2i+2 . Suppose indirectly that (6.13) fn−1=p2i+2 q2i+2 . SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 17 This is equivalent to the negation of the following condition: (6.14) {p2i+2 ≤dn+p2i+2 < p2i+3 +p2i+2 0≤cn−q2i+2 < q2i+3. However, the negation of (6.14) is dn∈ [0, p2i+3[(6.15) or cn∈ [q2i+2, q2i+3 +q2i+2[.(6.16) However, using q2i+3 =a2i+3q2i+2 +q2i+1 and 1 ≤j < a2i+3 it is easily seen that both (6.15) and (6.16) contradict (6.8). Thus the indirect assumption is false, and we have (6.17) fn−1=p2i+2 q2i+2 . Now (6.12) and (6.17) is just what we had to prove.  The proof of Theorem 2.4 is more complicated, so we split it into several lemmas. However, these lemmas may be interesting themselves, too. Recall that l:= 2i+ 1. Lemma 6.1. Suppose that sn=pcnqdn. Then (6.18) fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 is equivalent to (6.19) {0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1. Proof. Put sj=qcjpdjfor j∈N. By (6.18) we have cn+l=cn−lq2i+1 and dn+l=dn+lp2i+1 for l= 0, . . . , k −1. Thus, by Lemma 5.1, more precisely by (5.7) we have {0≤dn+lp2i+1 < p2i+2 for l= 0, . . . , k −1 p2i+1 ≤cn−lq2i+1 < q2i+2 +q2i+1 for l= 0, . . . , k −1. In fact this is a system of 2kinequalities, kof them containing cn, and the other kcontaining dn. It is easy to see that the solution of this is just (6.19).  18 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Lemma 6.2. Suppose that sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.20) fn−1=p2i+2 q2i+2 , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i+2 q2i+2 is equivalent to (6.21) {max(0, p2i+2 −kp2i+1)≤dn< p2i+2 −(k−1)p2i+1 max(kq2i+1, q2i+2)≤cn< q2i+2 +q2i+1. Proof. Using Lemma 5.1 and Lemma 6.1 it is easily seen that (6.20) is equivalent to (6.22)                          p2i+2 ≤dn+p2i+2 < p2i+2 +p2i+3 0≤cn−q2i+2 < q2i+3 0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1 p2i+2 ≤dn+kp2i+1 < p2i+2 +p2i+3 0≤cn−kq2i+1 < q2i+3 and this set of conditions clearly is equivalent to (6.21).  Lemma 6.3. Suppose that sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.23) fn−1=p2i,k−1 q2i,k−1 , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i,k−1 q2i,k−1 is equivalent to (6.24) {0≤dn< p2i kq2i+1 ≤cn< kq2i+1 +q2i. Proof. Using Lemma 5.1 and Lemma 6.1 it is easily seen that (6.23) is equivalent to (6.25)                          p2i,k−1≤dn+p2i,k−1< p2i,k 0≤cn−q2i,k−1< q2i+1 0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1 p2i,k−1≤dn+kp2i+1 < p2i,k 0≤cn−kq2i+1 < q2i+1 SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 19 and (using also (3.3)) this set of conditions is clearly equivalent to (6.24).  Lemma 6.4. Suppose that sn=pcnqdnand 1≤k < a2i+2 + 1. Then (6.26) fn−1=p2i,k−1 q2i,k−1 , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i,k q2i,k is equivalent to (6.27) {p2i≤dn< p2i+1 kq2i+1 ≤cn< kq2i+1 +q2i. Proof. Here we have to split the proof in two cases, depending on k < a2i+2 or k=a2i+2. If k < a2i+2 then using Lemma 5.1 and Lemma 6.1 it is easily seen that (6.26) is equivalent to (6.28)                          p2i,k−1≤dn+p2i,k−1< p2i,k 0≤cn−q2i,k−1< q2i+1 0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1 p2i,k ≤dn+kp2i+1 < p2i,k+1 0≤cn−kq2i+1 < q2i+1 and (using also (3.3)) this set of conditions is clearly equivalent to (6.27). If k=a2i+2 then the same argument applies, except that the last two conditions in (6.28) are replaced by (6.29) p2i+2 ≤dn+kp2i+1 < p2i+2 +p2i+3 0≤cn−kq2i+1 < q2i+3. However, this set of conditions will be equivalent to the same (6.27) as in the case k < a2i+2. Lemma 6.5. Suppose that sn=pcnqdnand 1≤k < a2i+2 + 1. Then (6.30) fn−1=p2i,k q2i,k , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i,k−1 q2i,k−1 is equivalent to (6.31) {0≤dn< p2i kq2i+1 +q2i≤cn<(k+ 1)q2i+1. 20 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Proof. Here we have to split the proof in two cases, depending on k < a2i+2 or k=a2i+2. If k < a2i+2 then using Lemma 5.1 and Lemma 6.1 it is easily seen that (6.26) is equivalent to (6.32)                          p2i,k ≤dn+p2i,k < p2i,k+1 0≤cn−q2i,k < q2i+1 0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1 p2i,k−1≤dn+kp2i+1 < p2i,k 0≤cn−kq2i+1 < q2i+1 and (using also (3.3)) this set of conditions is clearly equivalent to (6.31). If k=a2i+2 then the same argument applies, except that the first two conditions in (6.32) are replaced by (6.33) p2i+2 ≤dn+p2i+2 < p2i+2 +p2i+3 0≤cn−q2i+2 < q2i+3. However, this set of conditions will be equivalent to the same (6.31) as in the case k < a2i+2. Lemma 6.6. Suppose that sn=pcnqdnand 1≤k < a2i+2. Then (6.34) fn−1=p2i,k q2i,k , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i,k q2i,k is equivalent to (6.35) {p2i≤dn< p2i+1 kq2i+1 +q2i≤cn<(k+ 1)q2i+1. Remark. We mention that the case k=a2i+2 is just the case described by Lemma 6.2. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 21 Table 1. 1≤k < a2i+2 D1∅ D2{dn∈[0, p2i+1[ cn∈[kq2i+1,(k+ 1)q2i+1[ D3{dn∈[p2i+2 −kp2i+1, p2i+2 −(k−1)p2i+1[ cn∈[q2i+2, q2i+1 +q2i+2[ D4∅ Proof. Using Lemma 5.1 and Lemma 6.1 it is easily seen that (6.26) is equivalent to (6.36)                          p2i,k ≤dn+p2i,k < p2i,k+1 0≤cn−q2i,k < q2i+1 0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1 p2i,k ≤dn+kp2i+1 < p2i,k+1 0≤cn−kq2i+1 < q2i+1 and (using also (3.3)) this set of conditions is clearly equivalent to (6.35).  Lemma 6.7. Suppose that sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.37) fn−1=p2i+1 q2i+1 , fn=fn+1 =· · · =fn+k−1=p2i+1 q2i+1 , fn+k=p2i+1 q2i+1 . is equivalent to (dn, cn)∈D1∪D2∪D3∪D4, where the sets Diare given in Table 1, 2 and 3. Proof. By Lemma 6.1 we already know that (6.18) is equivalent to (6.19), that fn−1=p2i+1 q2i+1 is equivalent to (6.38) {0≤dn−p2i+1 < p2i+2 q2i+1 ≤cn+q2i+1 < q2i+1 +q2i+2. 22 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Table 2. k=a2i+2 D1{dn∈[p2i+2 −kp2i+1, p2i+1[ cn∈[kq2i+1, q2i+1 +q2i+2[ D2{dn∈[0, p2i+1[ cn∈[kq2i+1,(k+ 1)q2i+1[ D3{dn∈[p2i+2 −kp2i+1, p2i+2 −(k−1)p2i+1[ cn∈[q2i+2, q2i+1 +q2i+2[ D4{dn∈[0, p2i+2 −(k−1)p2i+1[ cn∈[q2i+2,(k+ 1)q2i+1[ Table 3. k=a2i+2 + 1 D1{dn∈[0, p2i+2 −(k−1)p2i+1[ cn∈[kq2i+1, q2i+1 +q2i+2[ D2{dn∈[0, p2i+2 −(k−1)q2i+1[ cn∈[kq2i+1,(k+ 1)q2i+1[ D3{dn∈[0, p2i+2 −(k−1)p2i+1[ cn∈[kq2i+1, q2i+1 +q2i+2[ D4{dn∈[0, p2i+2 −(k−1)p2i+1[ cn∈[kq2i+1, q2i+1 +q2i+2[ and that under the assumption sn+k−1=pcn−kq2i+1 qdn+kp2i+1 the statement fn+k=p2i+1 q2i+1 is equivalent to (6.39) {0≤dn+kp2i+1 < p2i+2 q2i+1 ≤cn−kq2i+1 < q2i+1 +q2i+2. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 23 Clearly, the necessary and sufficient condition for (6.37) is (6.19) and not (6.38) and not (6.39), however, this latter is equivalent to dn∈[0, p2i+2 −(k−1)p2i+1[(6.40) and cn∈[kq2i+1, q2i+1 +q2i+2[(6.41) and        dn∈]− ∞, p2i+1[∪[p2i+1 +p2i+2,∞[ or cn∈]− ∞,0[∪[q2i+2,∞[ (6.42) and        dn∈]− ∞,−kp2i+1[∪[p2i+2 −kp2i+1,∞[ or cn∈]− ∞,(k+ 1)q2i+1[∪[(k+ 1)q2i+1 +q2i+2,∞[. (6.43) The above system in fact leads to four systems of inequalities depending on which part of (6.42) and (6.43) is considered. We shall call the solution set of these systems by Difor i= 1,2,3,4, and the union of the solutions of these systems is the equivalent condition for (6.37). Depending on the value of kthese solutions may differ, and the corresponding solutions to the different possibilities for kare just those summarized in Table ??.  Proof of Theorem 2.4. To prove our theorem it is enough to show that the sets specified by the relations (6.21), (6.24), (6.27), (6.31) and (6.35) cover exactly the same possibilities for (cn, dn), as the set D1∪D2∪D3∪D4. We have to split our proof in three parts. If k < a2i+2 then (6.21) takes the form (6.44) {p2i+2 −kp2i+1 ≤dn< p2i+2 −(k−1)p2i+1 q2i+2 ≤cn< q2i+2 +q2i+1. This is just the same as D3. Further, in this case the sets specified in (6.24), (6.27), (6.31) and (6.35) give a pairwise disjoint union of the set D2. Taking in account that we also have D1=D4=∅our proof is finished. 24 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU If k=a2i+2 then (6.21) takes again the form (6.44). In this case sets Di are not pairwise disjoint, however, here it is also easy to see that the union of the pairwise disjoint sets specified by (6.21), (6.24), (6.27) and (6.31) is just the set D1∪D2∪D3∪D4, which proves our theorem for k=a2i+2. Finally, the case k=a2i+2 + 1 is the simplest, since in this case (6.21) take the form (6.45) {0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1. Further D2⊂D1=D3=D4shows that D1∪D2∪D3∪D4=D1, which is just the set specified by (6.45)  References [1] J.-H. Evertse, K. Gy˝ ory, C. Stewart, R. Tijdeman,S-unit equations and their applications, in: New Advances in Transcendence Theory, A. Baker (ed.), Cambridge University Press, 1988, 110–174. [2] A. Ya. Khinchin,Continued Fractions, University of Chicago Press, 1964. [3] C. Kimberling,Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997), 122–126. [4] N. G. Moshchevitin,Khintchines singular Diophantine systems and their applications, Uspekhi Mat. Nauk 65 (2010), 43-126. [5] O. Perron,Die Lehre von den Kettenbrchen, Chelsea Publishing Company, New York, 1950. [6] W. M. Schmidt,Diophantine Approximation, Lecture Notes in Mathematics 785, Springer, 1980. [7] T. N. Shorey, R. Tijdeman,Exponential Diophantine Equations, Cambridge University Press, Cambridge, 1986. [8] R. Tijdeman,On integers with many small prime factors, Compositio Math. 26 (1973), 319–330. [9] R. Tijdeman,On the maximal distance between integers composed of small primes, Compositio Math. 28 (1974), 159–162. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 25 A. B´ erczes Institute of Mathematics, University of Debrecen Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen H-4010 Debrecen, P.O. Box 12, Hungary E-mail address:[email protected] A. Dujella University of Zagreb, Department of Mathematics, Bijeniˇ cka cesta 30, 10000 Zagreb, Croatia E-mail address:[email protected] L. Hajdu Institute of Mathematics, University of Debrecen Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen H-4010 Debrecen, P.O. Box 12, Hungary E-mail address:[email protected]