arithmetic properties of integers with missing digits: distribution in residue classes
TRANSCRIPT
Periodica Mathematica Hungarica Vol. 42 (1–2), (2001), pp. 145–162
ARITHMETIC PROPERTIES OF INTEGERS WITHMISSING DIGITS: DISTRIBUTION IN RESIDUE CLASSES
Sergei Konyagin (Moscow)
Dedicated to Professor Andras Sarkozy on the occasion of his 60th birthday
Abstract
We show that numbers with missing digits are in average well-distributed inresidue classes modm where averaging is taken over m.
§1. Introduction
Let
(1) g ∈ N, g ≥ 3, t ∈ N, 2 ≤ t ≤ g − 1,
(2) D ⊂ {0, 1, . . . , g − 1}, 0 ∈ D, |D| = t,
and let WD(N) denote the set of integers n such that 0 ≤ n < N and representingn in the number system to base g:
n =ν−1∑j=0
ajgj, 0 ≤ aj ≤ g − 1,
where gν−1 < N ≤ gν , we have aj ∈ D for j = 0, . . . , ν. The arithmetic structureof the set WD(N) has been studied in [4], [2], [3]; further references are given in[3]. P. Erdos, C. Mauduit and A. Sarkozy [2] proved, among many other interestingresults, that the set WD(N) is well-distributed in the modulo m residue classes if(g,m) = 1 and m < exp(c(g, t)(logN)1/2). The objective of this paper is to continuestudy of distribution of WD(N) in residue classes.
Theorem 1. If g and t satisfy (1), (2) holds, and writing D = {d1, . . . , dt}where d1 = 0, we have (d2, . . . , dt) = 1, moreover, N ∈ N, ν0 ∈ N, N0 ≥ 0,N ≡ N0(mod gν0), K ∈ N, M ⊂ N, (g,m) = 1 for all m ∈ M, µ1, . . . , µK+1 are
Mathematics subject classification number: 11A63.Key words and phrases: digit properties, congruences.
0031-5303/01/$5.00 Akademiai Kiado, Budapestc© Akademiai Kiado, Budapest Kluwer Academic Publishers, Dordrecht
146 s. konyagin
integers, µ1 ≥ µ2 · · · ≥ µK > µK+1 = 0, Mk = gµk , and for any k = 1, . . . ,K + 1and any distinct elements m1, . . . ,mk from M we have (m1, . . . ,mk) ≤Mk, then∑m∈M
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣ ≤ |WD(N0)||M|M1
+|WD(N)|
K∑k=1
(1 + g
(1− 1
(g − 1)5(2g)2
)[ν0/2µk])2µk
−K
.
By our convention, in the case k > |M| we consider that the condition(m1, . . . ,mk) ≤Mk holds.
Theorem 1 yields a wider range of m’s for which the set WD(N) is well-distributed modulo m.
Corollary 1. If g and t satisfy (1), then there exist effectively computablepositive constants c1 = c1(g) > 0, c2 = c2(g) > 0, c3 = c3(g) > 0 such that if also(2) holds, and writing D = {d1, . . . , ds} where d1 = 0, we have (d2, . . . , dt) = 1,moreover, N ∈ N, N ≥ 3, m ∈ N, m ≥ 2, (g,m) = 1, m < exp(c1 logN/ log logN)and h ∈ Z, then ∣∣∣∣|{n : n ∈WD(N), n ≡ h(modm)}| − 1
m|WD(N)|
∣∣∣∣<c2m|WD(N)| exp
(−c3
logNlogm
).
Also, the set WD(N) is well-distributed modulo m for almost all moduli m ≤N c coprime to g where c > 0 depends only on g.
Corollary 2. If g and t satisfy (1), then there exist effectively computablepositive constants c4 = c4(g) > 0, c5 = c5(g) > 0, c6 = c6(g) > 0 such that if also(2) holds, and writing D = {d1, . . . , ds} where d1 = 0, we have (d2, . . . , dt) = 1,moreover, N ∈ N, N ≥ 3, M ∈ N, M ≤ N c4 , then∑
m≤M, (g,m)=1
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤ c5|WD(N)|M exp(−c6
√logN
).
Recently C. Dartyge and C. Mauduit [1] have established similar results; theirarguments differ from ours. In particular, they have proved that for D = {0, 1},M = X
π4m (1+o(1)) as m→∞ and any A > 0 there exists B > 0 so that∑
m≤M(logN)−B,(g,m(m−1))=1
maxa(modm)
∣∣∣∣|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|m
∣∣∣∣� |WD(N)|(logN)A
.
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Note that their bound for M is better than one following from the proof of Corol-lary 2.
If the elements of the set M are pairwise coprime, then we can take K = 1and Theorem 1 can be rewritten in the following form.
Corollary 3. If g and t satisfy (1), (2) holds, and writing D = {d1, . . . , dt}where d1 = 0, we have (d2, . . . , dt) = 1, moreover, N ∈ N, ν0 ∈ N, N0 ≥ 0,N ≡ N0(mod gν0), M ⊂ {1, . . . ,M}, (g,m) = 1 for all m ∈ M, elements of Mare pairwise coprime, µ ∈ N, M = gµ, then∑
m∈Mmax
a(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤ |WD(N0)||M|M + |WD(N)|
(1 + g
(1− 1
(g − 1)5(2g)2
)[ν0/2µ])2µ
− 1
.
Observe that if WD(N) does not meet some residue class modulo m, then
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣ ≥ |WD(N)|.
Taking in Corollary 3 N = gν , ν0 = ν, N0 = 0, we immediately get
Corollary 4. If g and t satisfy (1), (2) holds, and writing D = {d1, . . . , dt}where d1 = 0, we have (d2, . . . , dt) = 1, moreover, ν ∈ N, N = gν, µ ∈ N, M = gµ,M⊂ {1, . . . ,M}, (g,m) = 1 for all m ∈ M, elements of M are pairwise coprime,and for any m ∈ M the set WD(N) does not meet some residue class modulo m,then
|M| ≤(
1 + g
(1− 1
(g − 1)5(2g)2
)[ν/2µ])2µ
− 1.
We shall show that, in turn, Corollary 4 implies
Corollary 5. If g and t satisfy (1), (2) holds, and writing D = {d1, . . . , dt}where d1 = 0, we have (d2, . . . , dt) = 1, moreover, ν ∈ N, ν > 2, gν−1 < N ≤ gν,µ ∈ N, M = gµ, N ∈ N, M ⊂ {1, . . . ,M}, (g,m) = 1 for all m ∈ M, elements ofM are pairwise coprime,
|M| ≥(
1 + g
(1− 1
(g − 1)5(2g)2
)[(ν−2)/2µ])2µ
,
then there is n ∈WD(N), n > 0, divisible by some m ∈M.
It has been established in [2] that if t > g3/4, a positive integer z is smallenough in terms of t and g (namely, z < (2(1 − log t/ log g))−1), then WD(N)contains integers with zth power parts as large as cN c with c = c(g, t, z) > 0.Corollary 6 yields that the last property holds for all g, t and z.
148 s. konyagin
Corollary 6. Let the conditions (1) and (2) be satisfied. Then there existeffectively computable positive constants c7 = c7(g) > 0 and c8 = c8(g) > 0 suchthat if z ≥ 2 is a positive integer and N ≥ zc7z, then there are n ∈ WD(N) andprime p such that n > 0, p > N c8/(z log z) and pz|n.
The bound for N in Corollary 1 cannot be essentially improved.
Theorem 2. If g and t satisfy (1), (2) holds, then for some constant c9 =c9(g) there are sequences {Ni}, {mi}, {hi}, such that for all i = 1, 2, . . .
mi < exp(c9 logNi/ log logNi)
butmi|{n : n ∈WD(Ni), n ≡ hi(modmi)}|/|WD(Ni)| → ∞ (i→∞).
Problem. Does there exist a constant c = c(g) > 0 such that for any m ≤ N c
with (g,m) = 1 the set WD(N) meets all residue classes modulo m?
In §2 we shall prove Theorem 1. Corollaries 1, 2, 5, 6 will be proved in §3. In§4 we shall prove Theorem 2.
Throughout the paper by c with subscripts we shall denote positive numbersdepending only on g.
§2. Proof of Theorem 1
Consider first the case
(2.1) N0 = 0.
We write e(u) = e2πiu. For any m and a we have
(2.2) m |{n ∈WD(N) : n ≡ a(modm)}| =m−1∑b=0
e(−ab/m)∑
n∈WD(N)
e(bn/m).
Thus,
m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)| =m−1∑b=1
e(−ab/m)∑
n∈WD(N)
e(bn/m)
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣ ≤ m−1∑
b=1
|SN (b/m)|,
whereSN (u) =
∑n∈WD(N)
e(un),
arithmetic properties of integers 149
and, finally,
(2.3)
∑m∈M
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤∑m∈M
m−1∑b=1
|SN (b/m)|.
For any rational u ∈ (0, 1) by κ(u) denote the number of the pairs (b,m), m ∈ M,b ∈ {1, . . . ,m− 1} such that u = b/m. The inequality (2.3) can be rewritten as
(2.4)
∑m∈M
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤∑u
κ(u)|SN (u)| =∑k≥1
∑u:κ(u)≥k
|SN (u)|.
Now notice that for any u = b′/m′ ∈ (0, 1), (b′,m′) = 1 the inequality κ(u) ≥k implies that m′ divides at least k elements of M, and, therefore, m′ ≤ Mk.Moreover, κ(u) ≤ K for any u. It follows from (2.4) that
(2.5)
∑m∈M
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤K∑k=1
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<Mk
∣∣∣∣SN ( b
m
)∣∣∣∣ .The right-hand side of (2.5) can be estimated by the following lemma.
Lemma 1. If g and t satisfy (1), (2) holds, and writing D = {d1, . . . , dt} whered1 = 0, we have (d2, . . . , dt) = 1, moreover, N ∈ N, ν0 ∈ N, N ≡ 0(modgν0),µ ∈ N, M = gµ, then ∑
bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SN ( b
m
)∣∣∣∣
≤ |WD(N)|
(1 + g
(1− 1
(g − 1)5(2g)2
)[ν0/2µ])2µ
− 1
.
Clearly, in the case (2.1) Theorem 1 immediately follows from Lemma 1 andthe inequality (2.5).
Proof of Lemma 1. Denote r = [ν0/(2µ)], N1 = g2rµ. Since 2rµ ≤ ν0 andN is divisible by gν0 , the number N ′ = N/N1 is a positive integer. Any number
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n ∈WD(N) is uniquely represented as
n =r−1∑j=0
M2jnj +M2rn′, nj ∈WD(M2), n′ ∈WD(N ′).
Therefore, for any u
(2.6)
SN (u) =∑
n0,...,nr−1∈WD(M2),
n′∈WD(N ′)
e
ur−1∑j=0
M2jnj +M2rn′
=
r−1∏j=0
∑nj∈WD(M2)
e(M2junj)
∑n′∈WD(N ′)
e(M2run′)
=
r−1∏j=0
SM2(M2ju)
SN ′(M2ru).
The last sum SN ′(M2ru) can be estimated trivially:
|SN ′(M2ru)| ≤ |WD(N ′)| = |WD(N)|/|WD(N1)| = |WD(N)|/t2rµ.
Substituting this estimate into (2.6), we find
|SN(u)| ≤ |WD(N)|t2rµ
r−1∏j=0
|SM2(M2ju)|.
By the Holder inequality,
(2.7)
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SN ( b
m
)∣∣∣∣
≤ |WD(N)|t2rµ
r−1∏j=0
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SM2
(M2jb
m
)∣∣∣∣r
1/r
.
Observe that for any j the numbers bM2j run over the reduced residue systemmod m together with b. Thus, all the sums on the right-hand side of (2.7) are equal,and we get
(2.8)∑
bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SN ( b
m
)∣∣∣∣ ≤ |WD(N)|t2rµ
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SM2
(b
m
)∣∣∣∣r .
arithmetic properties of integers 151
Denote s(u) = Sg(u)/t =∑tk=1 e(dku)/t. Similarly to (2.6), we have
SM2(u) =2µ−1∏j=0
Sg(gju),
or,
SM2(u) = t2µ2µ−1∏j=0
s(gju).
Denote by nj the closest integer to gjb/m; if gjb/m − 1/2 is an integer, then weset nj = gjb/m− 1/2. Let J(b,m) = {j : 0 ≤ j < 2µ, |gjb/m− nj | ≥ 1/(2g)}. ByLemma 2 from [2] we have |Sg(bgj/m)| ≤ 1− 1
(g−1)5(2g)2 for j ∈ J . Therefore,∣∣∣∣SM2
(b
m
)∣∣∣∣ ≤ t2µ(1− 1(g − 1)5(2g)2
)|J(b,m)|,
and from the last inequality and (2.8) we obtain
(2.9)
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SN ( b
m
)∣∣∣∣ ≤ |WD(N)|
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
(1− 1
(g − 1)5(2g)2
)r|J(b,m)|.
We will prove that there are few pairs (b,m) for which J(b,m) is small; namely,
(2.10) {(b,m) : 0 < b < m, (b,m) = 1, (g,m) = 1,m < M, |J(b,m)| = 0} = ∅and for s > 0(2.11)
|{(b,m) : 0 < b < m, (b,m) = 1, (g,m) = 1,m < M, |J(b,m)| = s}| ≤ gs(
2µs
).
Denote ∆j = nj+1 − gnj . By the choice of nj , |∆j | ≤ g. Then j ∈ J(b,m)if and only if ∆j 6= 0. (Notice that we need not care about the “intermediate”case |gjb/m − nj | = 1/(2g); this equality can never hold because (g,m) = 1.)Assume that |J(b,m)| = 0. Then nj+1 = gnj for j = 0, . . . , 2µ− 1. Consequently,n2µ = g2µn0 = M2n0. The last equality can be rewritten as |M2b/m−M2n0| ≤ 1/2,or |b/m − n0| ≤ 1/(2M2). But this does not agree with the conditions 1 ≤ b ≤m − 1, m < M . So, (2.10) holds. Let us estimate the number of pairs (b,m) with|J(b,m)| = s > 0. There are
(2µs
)ways to choose J(b,m) and for any j ∈ J(b,m)
there are at most g ways to choose ∆j since |∆j | ≤ g/2 and ∆j 6= 0 for j ∈ J(b,m).Recall that ∆j = 0 for j 6∈ J(b,m). Thus, there are at most gs
(2µs
)ways to choose
the whole sequence {∆j} (j = 0, . . . , 2µ − 1). If we know this sequence and n0,then the sequence {nj} (j = 0, . . . , 2µ− 1) is also determined. Moreover, it is easyto express n0 in terms of {∆j}: if the first nonzero term of the sequence {∆j} is
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positive, then n0 = 0, and if the first nonzero term of this sequence is negative,then n0 = 1. So, we have checked that there are at most gs
(2µs
)ways to choose
the sequence {nj} (j = 0, . . . , 2µ). But at most one pair (b,m) can correspond to afixed value n2µ. Indeed, assume the contrary: there are two distinct numbers b/mand b′/m′ such that |M2b/m− nj | ≤ 1/2 and |M2b′/m′ − nj | ≤ 1/2. Then∣∣∣∣ bm − b′
m′
∣∣∣∣ ≤ 1M2
.
On the other hand, ∣∣∣∣ bm − b′
m′
∣∣∣∣ ≥ 1mm′
>1M2
.
Thus, our assumption cannot hold, and (2.11) is proved.
Now, from (2.9)–(2.11) we obtain
∑bm∈(0,1), (b,m)=1,
(g,m)=1, m<M
∣∣∣∣SN ( b
m
)∣∣∣∣ ≤ |WD(N)|2µ∑s=1
gs(
2µs
)(1− 1
(g − 1)5(2g)2
)rs
= |WD(N)|
(1 + g
(1− 1
(g − 1)5(2g)2
)[ν0/2µ])2µ
− 1
,
and Lemma 1 is proved.
To complete the proof of Theorem 1, we have to consider the case when thecondition (2.1) is not satisfied. Without loss of generality, we can assume that
(2.12) 0 < N0 < gν0 .
Denote N1 = N −N0. As we have proved,
(2.13)
∑m∈M
maxa(modm)
∣∣m|{n ∈WD(N1) : n ≡ a(modm)| − |WD(N1)|∣∣
≤ |WD(N1)|
K∑k=1
(1 + g
(1− 1
(g − 1)5(2g)2
)[ν/2µk])2µk
−K
.
Taking into account (2.12), we see that any number n ∈ WD(N) \WD(N1) has aform n = N1 + n0 with n0 ∈WD(N0). Therefore,
(2.14)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤∣∣m|{n ∈WD(N1) : n ≡ a(modm)| − |WD(N1)|
∣∣+m|WD(N0)|.
Recalling that m ≤ M1 for all m ∈ M and summarizing (2.14) over m ∈ M, wecomplete the proof of Theorem 1.
arithmetic properties of integers 153
§3. Proof of Corollaries
Denote
(3.1) c10 =1
(g − 1)5(2g)2.
Also, let us fix a positive constant c4 = c4(g) satisfying the conditions
(3.2) 1 + g(1− c10)1
4c4−2 ≤ g1/4,
(3.3) 4c4 < log 2/ log g.
To prove Corollaries 1 and 2, we need two lemmas.
Lemma 2. If N is a sufficiently large real number, ν0, m and µ are positivereal numbers,
(3.4) N < g2ν0+2,
(3.5) m < N c4 ,
(3.6) m ≥ gµ−1,then (
1 + g (1− c10)[ν0/2µ])2µ≤ (gm)1/2.
Proof. By (3.5) and (3.6),
(3.7) µ ≤ logm/ log g + 1
and further
(3.8) µ < c4 logN/ log g + 1.
On the other hand, by (3.4),
(3.9) ν0 ≥ logN/(2 log g)− 1.
It follows from (3.8) and (3.9) that for sufficiently large N we haveν02µ≥ 1
4c4− 1.
Taking into account (3.2), we get
1 + g (1− c10)[ν0/2µ] ≤ 1 + g(1− c10)1
4c4−2 ≤ g1/4.
Therefore, using (3.6), we find(1 + g (1− c10)[ν0/2µ]
)2µ≤ gµ/2 ≤ (gm)1/2,
and Lemma 2 is proved.
Now set
(3.10) c1 = c3 =c1010
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Lemma 3. If N is a sufficiently large real number, µ0, M and µ are positivereal numbers, the conditions (3.4) and (3.6) are satisfied and, moreover,
(3.11) m < exp(c1 logN/ log logN),
(3.12) m > exp(log1/3N),
then (1 + g(1− c10)[ν0/2µ]
)2µ− 1 ≤ 2 exp
(−c3
logNlogm
).
Proof. As N is large enough and m is large enough by (3.12), we have from(3.7) and (3.9)
(3.13) (1− c10)[ν0/2µ] ≤ (1− c10)0.2 logN/ logm.
Further,
(3.14)(1− c10)0.2 logN/ logm =
(e−0.1c10 logN/ logm
)2= e−c1 logN/ logme−c3 logN/ logm.
Moreover, by (3.11),
(3.15) e−c1 logN/ logm ≤ e−c1 logN/(c1 logN/ log logN) =1
logN.
From (3.13)–(3.15) we find
(1− c10)[ν0/2µ] ≤1
logNe−c3 logN/ logm.
Therefore, taking into account that, by (3.7) and (3.11), 2µglogN < 1 for large N , we
have (1 + g(1− c10)[ν0/2µ]
)2µ≤(
1 +g
logNe−c3 logN/ logm
)2µ
< exp(
2µglogN
e−c3 logN/ logm)< exp
(e−c3 logN/ logm
)< 1 + 2 exp (−c3 logN/ logm) .
This completes the proof of Lemma 3.
Proof of Corollary 1. Clearly, we may consider that N is large enough.Also, the required estimate was proved in [2] for m < exp(c11
√logN) and we can
(and we shall) assume that the inequality (3.12) holds. Take an integer µ so that
(3.16) gµ−1 ≤ m < M = gµ.
arithmetic properties of integers 155
To estimate |{n : n ∈WD(N), n ≡ h( mod m)}| for an arbitrary m satisfying (3.12),(3.16) and (g,m) = 1, let us take in Theorem 1 M = {m}, K = 1, µ1 = µ. Thenthe assertion of the theorem can be written as
(3.17)
∣∣m|{n ∈WD(N) : n ≡ h(modm)| − |WD(N)|∣∣ ≤ |WD(N0)|M
+|WD(N)|((
1 + g(1− c10)[ν0/2µ])2µ− 1).
Also, we take
(3.18) ν0 =[
logN2 log g
]and
(3.19) N0 < gν0 .
We define c1 and c3 by (3.10) and assume that (3.11) holds.
Let us estimate the first term on the right-hand side of (3.17). From (3.18)and (3.19) we get
(3.20) WD(N) ≥ t2ν0 , WD(N0) ≤ tν0 .
For sufficiently large N from (3.11) and (3.12) we have
M exp(c3
logNlogm
)< N log 2/(2 log g).
Also, by (3.18), the inequality (3.4) holds. Thus,
M exp(c3
logNlogm
)< g(2ν0+2) log 2/(2 log g) = 2ν0+1 ≤ tν0+1,
and using (3.20) we obtain
(3.21) |WD(N0)|M ≤ t|WD(N)| exp(−c3
logNlogm
).
Now observe that (3.16) yields (3.6), and we can use Lemma 3 for estimationof the second term on the right-hand side of (3.17):
|WD(N)|((
1 + g(1− c10)[ν0/2µ])2µ− 1)≤ 2|WD(N)| exp
(−c3
logNlogm
).
Finally, substituting (3.21) and the last inequality into (3.17), we complete the proofof Corollary 1.
Proof of Corollary 2. We shall again consider that N is large enough.Also, we consider that
(3.22) M ≤ N c4 ,
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where c4 > 0 satisfies inequalities (3.2) and (3.3). Let us take in Theorem 1M = {m ≤ M : (g,m) = 1}, K = M . For any k ≤ K and any distinct ele-ments m1, . . . ,mk fromM we have (m1, . . . ,mk) ≤M/k. Therefore, the condition(m1, . . . ,mk) ≤Mk holds for
(3.23) Mk = gµk , µk = [log(M/k)/ log g] + 1.
The statement of the theorem can be written as
(3.24)
∑m≤M, (g,m)=1
maxa(modm)
∣∣m|{n ∈WD(N) : n ≡ a(modm)| − |WD(N)|∣∣
≤ |WD(N0)||M|M1
+|WD(N)|(
K∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1))
.
As in the proof of Corollary 1, we take ν0 and N0 by (3.18) and (3.19). As we haveseen, these conditions imply (3.4) and (3.20).
Let us estimate the first term on the right-hand side of (3.24). Note that, by(3.22), (3.4), and (3.3),
MN c4 ≤ N2c4 ≤ g2c4(2ν0+2) = g4c4g4c4ν0
≤ g4c42ν0 ≤ g4c4tν0 .From the last inequality, using (3.20), (3.23) and again (3.20), we get
(3.25)|WD(N0)||M|M1 ≤ tν0M(gM) ≤ tν0g4c4tν0N−c4(gM)
= g1+4c4t2ν0MN−c4 ≤ g1+4c4WD(N)MN−c4 .
To estimate the second term on the right-hand side of (3.24), we split the sumover k into two parts:
(3.26)
K∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)
=K1∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)
+K∑
k=K1+1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1),
where K1 is chosen from the inequalities
(3.27)M
K1 + 1≤ exp(
√logN) <
M
K1
.
Take an arbitrary k ≥ 1. We can use Lemma 2 for m = M/k, µ = µk:(1 + g (1− c10)[ν0/2µk]
)2µk≤ gµk/2 ≤ (gM/k)1/2.
arithmetic properties of integers 157
Now we are ready to estimate the first sum on the right-hand side of (3.26):
K1∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)
≤K1∑k=1
(gM/k)1/2 ≤ c12(MK1)1/2,
and, by (3.27),
(3.28)K1∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)≤ c12M exp(−
√logN/2).
Let us estimate the second sum on the right-hand side of (3.26). Denoteµ = µK1
, m = M/K1. For any k > K1 we have(1 + g (1− c10)[ν0/2µk]
)2µk− 1 ≤
(1 + g (1− c10)[ν0/2µ]
)2µ− 1.
By Lemma 3,(1 + g (1− c10)[ν0/2µ]
)2µ− 1 ≤ 2 exp (−c3 logN/ logm) .
For sufficiently large N we have m ≤ exp(2√
logN). Thus,(1 + g (1− c10)[ν0/2µ]
)2µ− 1 ≤ 2 exp
(−c3
2
√logN
).
Taking the sum over k = K1 + 1, . . . ,K, we get
(3.29)K∑
k=K1+1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)≤ 2M exp
(−c3
2
√logN
).
Denotec6 = min(1/2, c3/2).
The bounds (3.28) and (3.29) yield
(3.30)K∑k=1
((1 + g (1− c10)[ν0/2µk]
)2µk− 1)≤ c13M exp
(−c6
√logN
).
Observing that N−c4 ≤ exp(−c6√
logN)
for sufficiently large N and substi-tuting the estimates (3.25) and (3.30) into (3.24), we complete the proof of Corol-lary 2.
Proof of Corollary 5. Let us apply Corollary 4 with ν and N replacedby ν′ = ν − 2 and N ′ = gν−2 respectively. We obtain that for some m ∈ M theset WD(N ′) meets all the residue classes modulo m. In particular, there existsn′ ∈ WD(N ′) such that n′ ≡ −dgν−2(modm) where d is an arbitrary nonzero
158 s. konyagin
element of D. Take n = n′ + dgν−2. Clearly, we have n ∈ WD(N), n > 0 andn ≡ 0(modm) as required.
Proof of Corollary 6. Define
(3.31) c14 = 3 + log g/ log 2
and
(3.32) c15 =c105c14
.
By the Prime Number Theorem, there exists u0 so that
(3.33) ∀u ≥ u0 π(u)− π(u/2) ≥ √gu.
Let
(3.34) c16 = max(
1c15
,log u0c15 log g
).
Now take a positive integer N and a corresponding positive integer ν: gν−1 <N ≤ gν . Assume that N is so large that
(3.35) ν − 2 ≥ c16z log z.
LetM =
{pz : g[2c15(ν−2)/(z log z)]/2 < p ≤ g[2c15(ν−2)/(z log z)]
},
where p denotes an arbitrary prime. Using Corollary 5, we shall prove that thereis a nonzero element n ∈ WD(N) divisible by some m ∈ M. This immediatelygives Corollary 6 since it is easy to find constants c7 and c8 such that the inequalityN ≥ zc7z implies (3.35) and for any p, pz ∈ M we have
p > g[2c15(ν−2)/(z log z)]/2 ≥ gc8ν/(z log z) ≥ N c8/(z log z).
By (3.35) and (3.34),
(3.36) c15(ν − 2)/(z log z) ≥ c15c16 ≥ 1.
Therefore,g[2c15(ν−2)/(z log z)]/2 ≥ g2/2 > g,
and all elements of M are coprime with g.Now we estimate the cardinality of M. Let u = g[2c15(ν−2)/(z log z)]. We have
(3.37) |M| = π(u)− π(u/2).
Moreover, u ≥ u0. Indeed, by (3.36), u ≥ gc15(ν−2)/(z log z). Further, by (3.35) and(3.34),
gc15(ν−2)/(z log z) ≥ gc15c16 ≥ u0.Therefore, by (3.37) and (3.33),
(3.38) |M| ≥ √gu > gc15(ν−2)/(z log z).
arithmetic properties of integers 159
Denote µ = [2c15(ν−2)/(z log z)]z, M = gµ. Clearly, m ≤M for any m ∈M.We have
(3.39) (ν − 2)/2µ ≥ ν − 22(2c15(ν − 2)z/(z log z))
= log z/4c15.
Subsequently using (3.1), (3.31), (3.32), we get
c10 < 0.001, c14 > 4, c15 < 0.00005.
Hence, log z/4c15 > 3000 and, by (3.39),
[(ν − 2)/2µ] ≥ [log z/4c15] > log z/5c15.
Further, by (3.32),
(3.40)
(1− 1
(g − 1)5(2g)2
)[(ν−2)/2µ]< exp (−c10[(ν − 2)/2µ])
< exp (−c10 log z/5c15) = z−c14 .
The relation (3.31) implies c14 ≥ 3 + log g/ log z and
(3.41) 1 + gz−c14 ≤ 1 + z−3 ≤ 1 + 1/4z < exp(1/4z) < g1/4z.
Combining (3.40), (3.41), and (3.39), we get(1 + g
(1− 1
(g − 1)5(2g)2
)[(ν−2)/2µ])2µ
< g2µ/4z ≤ gc15(ν−2)/(z log z).
So, by (3.38),
|M| ≥(
1 + g
(1− 1
(g − 1)5(2g)2
)[(ν−2)/2µ])2µ
,
all requirements of Corollary 5 hold, and there is indeed a nonzero element n ∈WD(N) divisible by some m ∈M. This completes the proof of Corollary 6.
§4. Proof of Theorem 2
As in the proof of Theorem 1, denote
SN (u) =∑
n∈WD(N)
e(un).
In particular, Sg(u) =∑d∈D e(ud). If d ∈ D, then∑−g/2≤b<g/2
Sg(b/g)e(−bd/g) = g.
Since Sg(0) = t < g, there exists b0 such that 0 < |b0| ≤ g/2 and
(4.1) Sg(b0/g) 6= 0.
160 s. konyagin
Moreover, if b0 < 0, then b0 can be replaced by −b0; thus, we can consider that
(4.2) 0 < b0 ≤ g/2.
Assume that i is large enough and take µ = µi = i, r = ri = [c17 log i], where c17will be fixed at the end of the proof, m = mi = gµ − 1, M = m+ 1, N = Ni = grµ.Clearly, the required inequality
mi < exp(c9 logNi/ log logNi)
holds for some constant c9 (for example c9 = 2/c17), and we have to check that forsome hi
(4.3) mi|{n : n ∈WD(Ni), n ≡ hi(modmi)}|/|WD(Ni)| → ∞ (i→∞).
It follows from (2.2) that
mm−1∑a=0
|{n ∈WD(N) : n ≡ a(modm)}|2 =m−1∑b=0
|SN (b/m)|2.
Therefore, for some h = hi
(4.4) m2 |{n ∈WD(N) : n ≡ h(modm)}|2 ≥m−1∑b=0
|SN (b/m)|2.
Similarly to (2.6), we can write the identity
SN (b/m) =r−1∏j=0
SM (M jb/m).
But, for any j and b we have M jb ≡ b(modm). Hence,
(4.5) SN (b/m) = (SM (b/m))r.
Also,
(4.6) SM (b/m) =µ−1∏j=0
Sg(gjb/m).
Notice that bgµ = bM ≡ b(modm); therefore,
(4.7) SM (gb/m) = SN(b/m).
Now let us estimate |SM (b0/m)| from below. We have gµi−1b0/mi → b0/g as i→∞.Hence, by (4.1), there is a constant c18 such that for sufficiently large i
(4.8) |Sg(gµ−1b0/m)| ≥ c18.
Assume that µ ≥ 2 and take arbitrary j ≤ µ − 2. Taking into account (4.2), wehave
gjb0/m ≤ gj+1/(2m) ≤ gj+1/2(gµ − gµ−2) = gj+3−µ/(2g2 − 2).
arithmetic properties of integers 161
Further, for any d ∈ D∣∣∣∣(d− g − 12
)gjb0/m
∣∣∣∣ ≤ gj+3−µ(g − 1)/(4g2 − 4) = gj+3−µ/(4g + 4) < 1/4
and
<(e(gjb0d/m)e(−(g − 1)gjb0/(2m))
)= <
(e
((d− g − 1
2
)gjb0/m
))≥ cos
(πgj+3−µ/(2g + 2)
).
Therefore,
(4.9)
|Sg(gjb0/m)| ≥ <(Sg(gjb0/m)e(−(g − 1)gjb0/2m)
)=∑d∈D<(e(gjb0d/m)e(−(g − 1)gjb0/(2m))
)≥ t cos
(πgj+3−µ/(2g + 2)
).
Combining (4.6), (4.8) and (4.9), we find
|SM (b0/m)| ≥ tµ∞∏l=0
cos(πg1−l/(2g + 2)
)c18/t ≥ c19tµ
with appropriate c19. By (4.7), we see that for any j
|SM(b0gj/m)| ≥ c19tµ.
Using (4.5), we get
|SN (b0gj/m)| ≥ cr19trµ = cr19|WD(N)|.
Now we are ready to estimate the right-hand side of (4.4):
m−1∑b=0
|SN(b/m)|2 ≥µ−1∑j=0
|SN (b0gj/m)|2 ≥ c2r19µ|WD(N)|2.
If c17 is small enough then c2r19 > µ−1/2, and (4.4) gives
m2 |{n ∈WD(N) : n ≡ h(modm)}|2 ≥ µ1/2|WD(N)|2.
So, (4.3) holds, and Theorem 2 is proved.
Acknowledgement. I wish to thank Professor A. Sarkozy and ProfessorC. Dartyge for useful suggestions.
The work on this paper was initiated in July 1999 when the author attendedthe “Paul Erdos and his mathematics” conference in Budapest and its satelliteworkshop on combinatorial number theory. My visit was supported by the Erdoscenter. It is my pleasure to thank the Erdos center for its hospitality and theexcellent working environment.
162 s. konyagin
REFERENCES
[1] C. Dartyge and C. Mauduit, Nombres presque premiers dont l’ecriture en baser ne comporte pas certains chiffres, to appear in J. Number Theory.
[2] P. Erdos, C. Mauduit and A. Sarkozy, On arithmetic properties of integerswith missing digits. I: Distribution in residue classes, J. Number Theory 70 (1998),99–120.
[3] P. Erdos, C. Mauduit and A. Sarkozy, On arithmetic properties of integerswith missing digits. II: Prime factors, Discrete Math. 200 (1999), 149–164.
[4] M. Filaseta and S. V. Konyagin, Squarefree values of polynomials all of whosecoefficients are 0 and 1, Acta Arith. 74 (1996), 191–205.
(Received: January 27, 2000)
S. KonyaginDepartment of Mechanics and MathematicsMoscow State UniversityMoscow, 119899RussiaE-mail: [email protected]