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 Abs ac . We p o e some diophan ine p ope ies o he sequence o S-uni s. 1. In oduc ion In ege s ha ing no p ime ac o s ou side a fixed se o p imes play im- po an ole and a e hea ily in es iga ed in se e al pa s o numbe heo y. Fo example, hey play special ole in diophan ine numbe heo y; see e.g. he classical su ey pape o E e se, Gy˝o y, S ewa and Tijdeman [1] o Chap e 1 o he book o Sho ey and Tijdeman [7] and he e e ences gi en he e. Fu he , he sequence o med o such in ege s is also o in e es . To be p ecise, fix p imes p1<· · · <p , and w i e sn o he sequence o in ege s composed o hese p imes, a anged in an inc easing o de . Tijdeman [8] and [9] p o ided sha p uppe and lowe bounds o he gaps be ween consecu i e e ms o he sequence, espec i ely. These bounds ha e he nice p ope y ha hey a e ”almos ” equal. Namely, Tijdeman p o ed ha (1.1) sn (log sn)c1< sn+1 −sn<sn (log sn)c2 hold wi h some effec i ely compu able absolu e cons an s c1and c2 o all index nwhich is la ge enough. In he p oo s o bo h he lowe and he uppe 2010 Ma hema ics Subjec Classi ica ion: P ima y 11B83; Seconda y: 11N25, 11J70. Keywo ds and Ph ases: S-uni , in ege s di isible by ixed p imes, con inued ac ion. The esea ch was suppo ed in pa by g an s K67580, K75566, K100339 (A.B., L.H.) and NK101680 (L.H.) o he Hunga ian Na ional Founda ion o Scien i ic Resea ch. The wo k is suppo ed by he T´ AMOP-4.2.2.C-11/1/KONV-2012-0001 p ojec . The p ojec is implemen ed h ough he New Hunga y De elopmen Plan, co- inanced by he Eu opean Social Fund and he Eu opean Regional De elopmen Fund. (A.B., L.H.). 1 2 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU bound in (1.1) he app oxima ion p ope ies o he uple (log p1, . . . , log p ) play a c ucial ole. These a e mainly used h ough Bake ’s heo y, bu in es ablishing he uppe bound also he con inued ac ions o log pi/log pj play a i al ole. In his pape we de elop a me hod o explici ly gi e he gaps in he sequence sn. In o he wo ds, o any e m snwe can find bo h sn−1and sn+1, a leas in p inciple, wi hou enume a ing all e ms o he sequence. Again, he e he app oxima ion p ope ies o he uple (log p1, . . . , log p ) a e decisi e. In he case when he e a e wo fixed p imes, we e en gi e an efficien and algo i hm o find hese e ms explici ly. This is done by he ca e ul analysis o he beha io o he con inued ac ions o log p1/log p2. Since o explain ou esul s and me hods in de ail we need se e al no ions and no a ion, we shall do ha in he nex sec ion. 2. Main esul s Le S={p1,p2, . . . , p }be a se o a ional p imes, and in he sequel suppose ha p1<p2<· · · <p . The ing o a ional S-in ege s is deno ed by ZS, and i s uni g oup by Z∗ S. Conside hose S-uni s, which a e na u al numbe s, and deno e by (sn) he sequence consis ing o hese numbe s in inc easing o de . Clea ly, any elemen o he sequence (sn) can be w i en in he o m sn=pcn,1 1pcn,2 2. . . pcn, wi h cn,i ∈Z≥0. Conside he hype plane P ⊂ R defined by P:= {(x1, . . . , x ) : x1log p1+· · · +x log p = 0}. Then Pis a subspace o R , in pa icula , i clea ly con ains he o igin. Fo a poin a= (a1, . . . , a )∈Z ≥0deno e by d(a) he Euclidean dis ance o he poin a om he hype plane Pin R . Theo em 2.1. The ollowing s a emen s a e ue: (i) Fo all a= (a1, . . . , a ),b= (b1, . . . , b )∈Z ≥0we ha e pa1 1. . . pa <pb1 1. . . pb ⇐⇒ d(a)< d(b). In pa icula , d(a) = d(b)i and only i a=b. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 3 (ii) Le ∈R>0, and w i e c( ) := log √log2p1+· · · + log2p . Then he smalles sn o which sn> is ha sn=pa1 1. . . pa wi h a= (a1, . . . , a )∈Z ≥0 o which o e e y b= (b1, . . . , b )∈Z ≥0 wi h d(b)> c( )we ha e c( )< d(a)< d(b). Simila ly, he la ges sn o which sn< is ha sn=pa1 1. . . pa wi h a= (a1, . . . , a )∈Z ≥0 o which o e e y b= (b1, . . . , b )∈Z ≥0 wi h d(b)< c( )we ha e c( )> d(a)> d(b). Fu he , in bo h cases acan be effec i ely de e mined. Rema k. The p oo o Theo em 2.1 is based upon some p ope ies o a ce ain special mul idimensional diophan ine app oxima ion. Fo he heo y o mul idimensional diophan ine app oxima ions o diffe en ypes see he excellen su ey pape o Moshce i in [4], and he e e ences gi en he e. In he special case = 2 we can o mula e much mo e p ecise esul s. In o de o do so, we need o in oduce some u he no a ion. F om now on le S={p,q}be a se o wo a ional p imes wi h p<q. Now he sequence (sn) may be w i en in he o m sn=pcnqdnwi h cn, dn∈Z≥0. We define he companion sequence ( n) o (sn) by (2.1) n:= dn+1 −dn cn−cn+1 . La e we shall p o e ha he elemen s o he sequence ( n) a e always well defined (i.e. cn−cn+1 = 0), hey a e always in lowes e ms (i.e. gcd(dn+1 −dn, cn−cn+1) = 1), and n≥0, wi h equali y p ecisely o alues o n o which sn<q. In he s a emen o ou esul s below we use no ions ela ed o he con- inued ac ions o eal numbe s. He e we use hese no ions wi hou any e e ence, howe e he concep s and esul s connec ed o con inued ac- ions which a e needed in he pape , a e summa ized in Sec ion 3. 4 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Gi en a conc e e elemen o he sequence (sn), he ollowing heo em gi es a simple algo i hm how o de e mine he nex elemen in he sequence. Theo em 2.2. Le he sequences (sn),(cn),(dn)and ( n)ha e he same meaning as abo e. Suppose ha we a e gi en sk=pckqdk. Then we can compu e sk+1 in he ollowing way: •Le u1 1be he uppe con e gen o log p log qwi h maximal denomina o o which 1≤ckholds. •Le u2 2be he lowe con e gen o log p log qwi h maximal nume a o o which u2≤dkholds. •Pu x:= | 1log p−u1log q| − | 2log p−u2log q|and (2.2) ck+1 =   ck− 1i x < 0, ck+ 2i x > 0,dk+1 =   dk+u1i x < 0, dk−u2i x > 0. Then we ha e sk+1 =pck+1 qdk+1 . Rema k. In iew o he me hod o he p oo , ha ing skone can explici ly gi e he e m sk−1o he sequence, simila ly o he e m sk+1. Howe e , since in he ligh o Theo em 2.2 his can be done in he ob ious way, we omi he de ails. In he ollowing heo em we summa ize basic p ope ies o he companion sequence, which sequence desc ibes how he exponen s o pand qchange when we mo e om sn o sn+1. Theo em 2.3. Le he sequences (sn),(cn),(dn)and ( n)ha e he same meaning as abo e. Then we ha e he ollowing p ope ies: (i) The sequence ( n)is well-defined, i.e. cn+1 =cn o all n∈N. (ii) We ha e n≥0 o all n∈N, wi h equali y p ecisely o hose alues o n o which sn<q. (iii) All companion ac ions na e con e gen s o log p log q, and •i nis an uppe con e gen hen ck+1 < ckand dk+1 > dk, •i nis a lowe con e gen hen ck+1 > ckand dk+1 < dk. (i ) Suppose ha he smalles index nsuch ha n=u is k. Then •i u is an uppe con e gen o log p log q hen we ha e sk=p and sk+1 =qu; SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 5 •i u is a lowe con e gen o log p log q hen we ha e sk=quand sk+1 =p . Con e sely, •i sk=p and sk+1 =qu hen k=u is an uppe con e gen o log p log qand kis he index o u in he sequence ( n); •i sk=quand sk+1 =p hen k=u is a lowe con e gen o log p log qand kis he index o u in he sequence ( n). ( ) Le pi,j qi,j be a con e gen o log p log q. The numbe o occu ences o pi,j qi,j in he sequence ( n)is exac ly pi+1qi+1, whe e pi+1 qi+1 is he p incipal con e gen o log p log q ollowing he p incipal con e gen pi qi=pi,0 qi,0. To unde s and well he s uc u e o ou sequence (sn) we need o know how he co esponding companion sequence ( n) beha es. Some o he mos impo an a ising ques ions a e he ollowing: •i we know he alue o n hen which alues can be aken by n−1 and n+1 espec i ely •how many consecu i e elemen s o he sequence nmay ha e he same alue pi,j qi,j . Theo ems 2.4 and 2.5 gi e a p ecise answe o hese ques ions. In one hand we p o e ha an in e media e con e gen canno be he alue o wo con- secu i e elemen s o ( n), and ha he e a e a mos aj+2 + 1 consecu i e elemen s o ( n) which assume he same alue pj qj. Fu he ou Theo ems desc ibe all possible pa e ns o med by exac ly k(1 ≤k≤aj+2 + 1) con- secu i e elemen s o ( n) assuming he same alue pj qj, and by he p eceding and he ollowing elemen s. Mo eo e , ou Lemmas in Sec ion 6 gi e neces- sa y and sufficien condi ions o cn= o dqsnand dn= o dpsnso ha sn−1 is he s a ing poin o such a conc e e pa e n. In he ollowing Theo em 2.4 we answe he abo e ques ion o p incipal con e gen s, and in Theo em 2.5 we do he same o in e media e con e - gen s. Theo em 2.4. Le us suppose ha in he sequence o companion ac ions we ha e he ollowing pa e n: (2.3) n−1=pl ql , n= n+1 =· · · = n+k−1=pl ql , n+k=pl ql . 6 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Then we ha e 1≤k≤al+1 + 1, and o ( n−1, n+k)we ha e he ollowing possibili ies: (i) I 1≤k < al+1 hen (2.4) ( n−1, n+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) I k=al+1 hen (2.5) ( n−1, n+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) I k=al+1 + 1 hen (2.6) ( n−1, n+k) = (pl+1 ql+1 ,pl+1 ql+1 ). Theo em 2.5. Suppose ha n=pl,j ql,j wi h some 1≤j < al+2 (i.e. nis an in e media e con e gen ). Then we ha e (2.7) n−1= n+1 =pl+1 ql+1 . 3. Con inued ac ions In his sec ion we summa ize impo an p ope ies o he con inued ac- ion expansion and he co esponding con e gen s o eal numbe s. Fo he gene al heo y o con inued ac ions we e e o he classical books [2], [5], [6] and he e e ences gi en he e. I S={p,q}, hen, as we ha e seen, he s uc u e o he sequence o na u al S-uni s is s ongly connec ed o he con e gen s o he eal numbe log p log q. The p oo s o he p ope ies lis ed below may be ound in [2], [5] and [6]. Le 0 =α∈Rbe a eal numbe and define a0, a1, a2, . . . in he ollowing way: α0:= α,a0:= [α0], αi+1 := {1 {αi}},ai+1 := [ 1 αi+1 ], . The sequence (an) is called he con inued ac ion o α. In he sequel, o 0 =α∈Rwe shall deno e by [a0, a1, a2...] he con inued ac ion expansion o α. Pu (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 ac ions pi/qi o i≥0 a e called he p incipal con e gen s o α. Fu he , o non-nega i e in ege s iand jpu (3.3) pi,j =jpi+1 +pi, qi,j =jqi+1 +qi. The ac ions (3.4) pi,j qi,j =jpi+1 +pi jqi+1 +qi 1≤j≤ai+2 −1 a e called he in e media e con e gen s o α. We men ion, ha in many cases i is com o able o le in (3.4) he index jassume also he alues 0 and ai+2, in hese cases he esul ing ac ion in (3.4) being a p incipal con e gen , namely: (3.5) pi,0 qi,0 =pi qi and pi,ai+2 qi,ai+2 =pi+2 qi+2 . The p incipal con e gen s and in e media e con e gen s oge he a e called con e gen s. Fo he con e gen s o αwe ha e he ollowing p ope ies: · · · <pi qi <···<pi,j qi,j <pi,j+1 qi,j+1 <· · · <pi+2 qi+2 < . . . i iis e en,(3.6) · · · >pi qi >···>pi,j qi,j >pi,j+1 qi,j+1 >· · · >pi+2 qi+2 >· · · >i iis odd,(3.7) and pi,j−1qi,j −pi,jqi,j−1= (−1)j o i≥0 and 1 ≤j≤ai+2 −1. In he sequel he ac ions (3.6) o e en indices will also be e e ed o as lowe con e gen s, while he ac ions (3.7) o odd indices as uppe con e gen s. This e minology is clea ly jus ified by he ac , ha lowe con e gen s o αa e smalle hen α, while uppe con e gen s o αa e la ge hen α. We say ha • he a ional numbe p qis a bes app oxima ion o αi o e e y a ional numbe b cwi h denomina o c < q we ha e (3.8) |qα −p|<|cα −b| 8 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU • he a ional numbe p qis a bes lowe app oxima ion o αi p q< α and o e e y a ional numbe b c< α wi h denomina o c<qwe ha e (3.9) qα −p < cα −b • he a ional numbe p qis a bes uppe app oxima ion o αi p q> α and o e e y a ional numbe b c> α wi h denomina o c<qwe ha e (3.10) p−qα < b −cα. The fi s s a emen o he ollowing lemma is a well-known p ope y o p incipal con e gen s (see e.g. [2], [5], [6]), while he second and hi d s a emen s a e due o Kimbe ling [3]. Lemma 3.1. Le α= 0 be a eal numbe , and deno e by pi/qi o i≥0 he p incipal con e gen s o αand by pi,j qi,j o i≥0,1≤j < ai+2 he in e media e con e gen s o α. Then he ollowing s a emen s a e ue: (i) I b csa isfies |cα −b|<|qiα−pi| hen c≥qi+1 (ii) The bes lowe app oxima es o αa e he lowe con e gen s o α, i.e. he ac ions pi,j qi,j o e en iand 0≤j < ai+2. (iii) The bes uppe app oxima es o αa e he uppe con e gen s o α, i.e. he ac ions pi,j qi,j o odd iand 0≤j < ai+2. Rema k. The fi s s a emen o Lemma 3.1 implies as a simple co olla y ha he bes app oxima es o αa e he p incipal con e gen s o α. In he las wo s a emen s o Lemma 3.1 among he bes lowe and uppe app oxima ions pi,j qi,j o 0 ≤j < ai+2 we can find he p incipal con e gen s, i.e. he ac ions wi h j= 0, and he in e media e con e gen s, i.e. he ac ions wi h 1 ≤j < ai+2. The nex lemma is a classical esul o con inued ac ions again (see e.g. [2], [5], [6]). Lemma 3.2. Suppose ha p q= 0 is a con e gen o a posi i e eal numbe α. Then q pis a con e gen o 1 α. The pa i y o he index o q pamong he con e gen s o 1 αis opposi e o he pa i y o he index o p qamong he con e gen s o α. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 9 4. Applica ions In his sec ion we gi e some diophan ine applica ions o ou esul s. Theo em 4.1. The e exis infini ely many indices ksuch ha he e ms sk, sk+1, sk+2, sk+3 o m a geome ic p og ession. P oo o Theo em 4.1. In ac we p o e mo e. Fi s no e ha since α:= log p log q is anscenden al, he con inued ac ion expansion o αcon ains infini ely many e ms >1, so he e a e ei he infini ely many odd alues o nwi h an+1 >1 o he e a e ei he infini ely many e en alues o nwi h an+1 >1. Fi s suppose ha he e a e infini ely many odd alues o nwi h an+1 >1 and ake a fixed odd index nsuch ha an+1 >1. Obse e ha hen we ha e qn+1 =an+1qn+qn−1≥2qn+qn−1, and pn+1 =an+1pn+pn−1≥2pn+pn−1. Choose in ege s Aand Bsubjec o he ollowing es ic ions: (4.1) 3qn≤A < 3qn+qn−1,0≤B < pn−1. We claim ha wi h any o he abo e choices o Aand B, w i ing sk= pAqBwe ha e k=pn qn, and he e ms sk, sk+1, sk+2, sk+3 o m a geome ic p og ession. To check hese asse ions, obse e ha bo h A < qn+qn+1 ≤min{qn+2, qn,1}and B+ 2pn< pn+1 holds. Hence by Theo em 2.2 we clea ly ge ha sk=pAqB, sk+1 =pA−qnqB+pn, sk+2 =pA−2qnqB+2pn, sk+3 =pA−3qnqB+3pn is a desi ed geome ic p og ession. Since by ou assump ion he e a e infin- i ely many indices nha ing he desi ed p ope y, he s a emen ollows. Now we also ha e o deal wi h he case when he e a e only fini ely many odd alues o nwi h an+1 >1. Howe e , in his case he e a e infini ely many e en alues o nwi h an+1 >1 and choosing any such na simila cons uc ion is possible as abo e, jus we ha e o choose Aand Bsubjec o he es ic ions 0≤A < qn−1,3pn≤B < 3pn+pn−1, 16 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU 6. P oo o Theo ems 2.4 and 2.5 In o de o p o e Theo em 2.4 and 2.5 we need o sepa a e he cases whe e lis odd and lis e en. He e we only p o e he case when lis odd and we men ion ha he o he case can be p o ed in he e y same way. Du ing he p oo s we shall use (3.3) se e al imes wi hou u he e e ence. Fo he es o his sec ion pu l:= 2i+ 1. Fi s we p o e Theo em 2.5, since i s p oo is much simple . P oo o Theo em 2.5. Lemma 5.1 shows ha i 1 ≤j < a2i+3 hen n= p2i+1,j q2i+1,j is equi alen o (6.8) {0≤dn< p2i+2 q2i+1,j ≤cn< q2i+1,j+1. Fu he , n=p2i+1,j q2i+1,j also yields cn+1 =cn−q2i+1,j and dn+1 =dn+p2i+1,j. These, oge he wi h (6.8) show ha we ha e (6.9) {p2i+1,j ≤dn+1 < p2i+2 +p2i+1,j 0≤cn+1 < q2i+1,j+1 −q2i+1,j, his la e being equi alen o (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 he consequence (6.11) {p2i+2 ≤dn+1 < p2i+3 +p2i+2 0≤cn+1 < q2i+3, which p o es (6.12) n+1 =p2i+2 q2i+2 . Now we p o e he s a emen n−1=p2i+2 q2i+2 . Suppose indi ec ly ha (6.13) n−1=p2i+2 q2i+2 . SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 17 This is equi alen o he nega ion o he ollowing condi ion: (6.14) {p2i+2 ≤dn+p2i+2 < p2i+3 +p2i+2 0≤cn−q2i+2 < q2i+3. Howe e , he nega ion o (6.14) is dn∈ [0, p2i+3[(6.15) o cn∈ [q2i+2, q2i+3 +q2i+2[.(6.16) Howe e , using q2i+3 =a2i+3q2i+2 +q2i+1 and 1 ≤j < a2i+3 i is easily seen ha bo h (6.15) and (6.16) con adic (6.8). Thus he indi ec assump ion is alse, and we ha e (6.17) n−1=p2i+2 q2i+2 . Now (6.12) and (6.17) is jus wha we had o p o e.  The p oo o Theo em 2.4 is mo e complica ed, so we spli i in o se e al lemmas. Howe e , hese lemmas may be in e es ing hemsel es, oo. Recall ha l:= 2i+ 1. Lemma 6.1. Suppose ha sn=pcnqdn. Then (6.18) n= n+1 =· · · = n+k−1=p2i+1 q2i+1 is equi alen o (6.19) {0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1. P oo . Pu sj=qcjpdj o j∈N. By (6.18) we ha e cn+l=cn−lq2i+1 and dn+l=dn+lp2i+1 o l= 0, . . . , k −1. Thus, by Lemma 5.1, mo e p ecisely by (5.7) we ha e {0≤dn+lp2i+1 < p2i+2 o l= 0, . . . , k −1 p2i+1 ≤cn−lq2i+1 < q2i+2 +q2i+1 o l= 0, . . . , k −1. In ac his is a sys em o 2kinequali ies, ko hem con aining cn, and he o he kcon aining dn. I is easy o see ha he solu ion o his is jus (6.19).  18 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU Lemma 6.2. Suppose ha sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.20) n−1=p2i+2 q2i+2 , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i+2 q2i+2 is equi alen o (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. P oo . Using Lemma 5.1 and Lemma 6.1 i is easily seen ha (6.20) is equi alen o (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 his se o condi ions clea ly is equi alen o (6.21).  Lemma 6.3. Suppose ha sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.23) n−1=p2i,k−1 q2i,k−1 , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i,k−1 q2i,k−1 is equi alen o (6.24) {0≤dn< p2i kq2i+1 ≤cn< kq2i+1 +q2i. P oo . Using Lemma 5.1 and Lemma 6.1 i is easily seen ha (6.23) is equi alen o (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)) his se o condi ions is clea ly equi alen o (6.24).  Lemma 6.4. Suppose ha sn=pcnqdnand 1≤k < a2i+2 + 1. Then (6.26) n−1=p2i,k−1 q2i,k−1 , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i,k q2i,k is equi alen o (6.27) {p2i≤dn< p2i+1 kq2i+1 ≤cn< kq2i+1 +q2i. P oo . He e we ha e o spli he p oo in wo cases, depending on k < a2i+2 o k=a2i+2. I k < a2i+2 hen using Lemma 5.1 and Lemma 6.1 i is easily seen ha (6.26) is equi alen o (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)) his se o condi ions is clea ly equi alen o (6.27). I k=a2i+2 hen he same a gumen applies, excep ha he las wo condi ions in (6.28) a e eplaced by (6.29) p2i+2 ≤dn+kp2i+1 < p2i+2 +p2i+3 0≤cn−kq2i+1 < q2i+3. Howe e , his se o condi ions will be equi alen o he same (6.27) as in he case k < a2i+2. Lemma 6.5. Suppose ha sn=pcnqdnand 1≤k < a2i+2 + 1. Then (6.30) n−1=p2i,k q2i,k , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i,k−1 q2i,k−1 is equi alen o (6.31) {0≤dn< p2i kq2i+1 +q2i≤cn<(k+ 1)q2i+1. 20 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU P oo . He e we ha e o spli he p oo in wo cases, depending on k < a2i+2 o k=a2i+2. I k < a2i+2 hen using Lemma 5.1 and Lemma 6.1 i is easily seen ha (6.26) is equi alen o (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)) his se o condi ions is clea ly equi alen o (6.31). I k=a2i+2 hen he same a gumen applies, excep ha he fi s wo condi ions in (6.32) a e eplaced by (6.33) p2i+2 ≤dn+p2i+2 < p2i+2 +p2i+3 0≤cn−q2i+2 < q2i+3. Howe e , his se o condi ions will be equi alen o he same (6.31) as in he case k < a2i+2. Lemma 6.6. Suppose ha sn=pcnqdnand 1≤k < a2i+2. Then (6.34) n−1=p2i,k q2i,k , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i,k q2i,k is equi alen o (6.35) {p2i≤dn< p2i+1 kq2i+1 +q2i≤cn<(k+ 1)q2i+1. Rema k. We men ion ha he case k=a2i+2 is jus he case desc ibed 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∅ P oo . Using Lemma 5.1 and Lemma 6.1 i is easily seen ha (6.26) is equi alen o (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)) his se o condi ions is clea ly equi alen o (6.35).  Lemma 6.7. Suppose ha sn=pcnqdnand 1≤k≤a2i+2 + 1. Then (6.37) n−1=p2i+1 q2i+1 , n= n+1 =· · · = n+k−1=p2i+1 q2i+1 , n+k=p2i+1 q2i+1 . is equi alen o (dn, cn)∈D1∪D2∪D3∪D4, whe e he se s Dia e gi en in Table 1, 2 and 3. P oo . By Lemma 6.1 we al eady know ha (6.18) is equi alen o (6.19), ha n−1=p2i+1 q2i+1 is equi alen o (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 ha unde he assump ion sn+k−1=pcn−kq2i+1 qdn+kp2i+1 he s a emen n+k=p2i+1 q2i+1 is equi alen o (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 Clea ly, he necessa y and sufficien condi ion o (6.37) is (6.19) and no (6.38) and no (6.39), howe e , his la e is equi alen o 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,∞[ o cn∈]− ∞,0[∪[q2i+2,∞[ (6.42) and        dn∈]− ∞,−kp2i+1[∪[p2i+2 −kp2i+1,∞[ o cn∈]− ∞,(k+ 1)q2i+1[∪[(k+ 1)q2i+1 +q2i+2,∞[. (6.43) The abo e sys em in ac leads o ou sys ems o inequali ies depending on which pa o (6.42) and (6.43) is conside ed. We shall call he solu ion se o hese sys ems by Di o i= 1,2,3,4, and he union o he solu ions o hese sys ems is he equi alen condi ion o (6.37). Depending on he alue o k hese solu ions may diffe , and he co esponding solu ions o he diffe en possibili ies o ka e jus hose summa ized in Table ??.  P oo o Theo em 2.4. To p o e ou heo em i is enough o show ha he se s specified by he ela ions (6.21), (6.24), (6.27), (6.31) and (6.35) co e exac ly he same possibili ies o (cn, dn), as he se D1∪D2∪D3∪D4. We ha e o spli ou p oo in h ee pa s. I k < a2i+2 hen (6.21) akes he o m (6.44) {p2i+2 −kp2i+1 ≤dn< p2i+2 −(k−1)p2i+1 q2i+2 ≤cn< q2i+2 +q2i+1. This is jus he same as D3. Fu he , in his case he se s specified in (6.24), (6.27), (6.31) and (6.35) gi e a pai wise disjoin union o he se D2. Taking in accoun ha we also ha e D1=D4=∅ou p oo is finished. 24 A. B´ ERCZES, A. DUJELLA, AND L. HAJDU I k=a2i+2 hen (6.21) akes again he o m (6.44). In his case se s Di a e no pai wise disjoin , howe e , he e i is also easy o see ha he union o he pai wise disjoin se s specified by (6.21), (6.24), (6.27) and (6.31) is jus he se D1∪D2∪D3∪D4, which p o es ou heo em o k=a2i+2. Finally, he case k=a2i+2 + 1 is he simples , since in his case (6.21) ake he o m (6.45) {0≤dn< p2i+2 −(k−1)p2i+1 kq2i+1 ≤cn< q2i+2 +q2i+1. Fu he D2⊂D1=D3=D4shows ha D1∪D2∪D3∪D4=D1, which is jus he se specified by (6.45)  Re e ences [1] J.-H. E e se, K. Gy˝ o y, C. S ewa , R. Tijdeman,S-uni equa ions and hei applica ions, in: New Ad ances in T anscendence Theo y, A. Bake (ed.), Camb idge Uni e si y P ess, 1988, 110–174. [2] A. Ya. Khinchin,Con inued F ac ions, Uni e si y o Chicago P ess, 1964. [3] C. Kimbe ling,Bes lowe and uppe app oxima es o i a ional numbe s, El- emen e de Ma hema ik, 52 (1997), 122–126. [4] N. G. Moshche i in,Khin chines singula Diophan ine sys ems and hei ap- plica ions, Uspekhi Ma . Nauk 65 (2010), 43-126. [5] O. Pe on,Die Leh e on den Ke enb chen, Chelsea Publishing Company, New Yo k, 1950. [6] W. M. Schmid ,Diophan ine App oxima ion, Lec u e No es in Ma hema ics 785, Sp inge , 1980. [7] T. N. Sho ey, R. Tijdeman,Exponen ial Diophan ine Equa ions, Camb idge Uni e si y P ess, Camb idge, 1986. [8] R. Tijdeman,On in ege s wi h many small p ime ac o s, Composi io Ma h. 26 (1973), 319–330. [9] R. Tijdeman,On he maximal dis ance be ween in ege s composed o small p imes, Composi io Ma h. 28 (1974), 159–162. SOME DIOPHANTINE PROPERTIES OF THE SEQUENCE OF S-UNITS 25 A. B´ e czes Ins i u e o Ma hema ics, Uni e si y o Deb ecen Numbe Theo y Resea ch G oup, Hunga ian Academy o Sciences and Uni e si y o Deb ecen H-4010 Deb ecen, P.O. Box 12, Hunga y E-mail add ess:[email p o ec ed] A. Dujella Uni e si y o Zag eb, Depa men o Ma hema ics, Bijeniˇ cka ces a 30, 10000 Zag eb, C oa ia E-mail add ess:[email p o ec ed] L. Hajdu Ins i u e o Ma hema ics, Uni e si y o Deb ecen Numbe Theo y Resea ch G oup, Hunga ian Academy o Sciences and Uni e si y o Deb ecen H-4010 Deb ecen, P.O. Box 12, Hunga y E-mail add ess:[email p o ec ed]