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.
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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)
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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)
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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)
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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)
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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)
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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
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dn∈]− ∞, p2i+1[∪[p2i+1 +p2i+2,∞[
o
cn∈]− ∞,0[∪[q2i+2,∞[
(6.42)
and
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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]