Abstract We prove several results which imply the following consequences. For any ε>0 and any sufficiently large prime p, if ℐ1,…,ℐ13 are intervals of cardinalities |ℐj|>p1/4+ε and abc≢0(modp), then the congruence ax1…x6+bx7…x13≡c(modp) has a solution with xj∈ℐj. There exists an absolute constant n0∈ℕ such that for any 0<ε<1 and any sufficiently large prime p, any quadratic residue λ modulo p can be represented in the form x1…xn0≡λ(modp),xi∈ℕ,xi≤p1/(4e2/3)+ε. For any ε>0, there exists n=n(ε)∈ℕ such that for any sufficiently large m∈ℕ the congruence x1…xn≡1(modm),xi∈ℕ,xi≤mε has a solution with x1≠1. 1. Introduction For a prime p, let Fp denote the field of residue classes modulo p and Fp* be the set of non-zero elements of Fp. We recall that a set I⊂Fp is called an interval if I={L+1,…,L+N}(modp) for some integers L and N≥1. Let I1,…,I2k be non-zero (that is, Ii≠{0}) intervals in Fp and let B be the box B=I1×I2×⋯×I2k. Given elements a,b∈Fp* and c∈Fp, we consider the equation ax1…xk+bxk+1…x2k=c;(x1,…,x2k)∈B. (1.1) The problem is to determine how large the size of the box B should be in order to guarantee the solvability of (1.1). The case k=2 was initiated in the work of Ayyad et al. [3], and then continued in [9, 2]. It was proved in [3] that there is a constant C such that if ∣B∣>Cp2log4p, then the equation ax1x2+bx3x4=c;(x1,x2,x3,x4)∈B (1.2) has a solution, and they asked whether the factor log4p can be removed. The authors of [9] relaxed the condition to ∣B∣>Cp2logp and also proved that (1.2) has a solution in any box B with ∣I1∣∣I3∣>15p and ∣I2∣∣I4∣>15p. The main question for k=2 was solved by Bourgain (unpublished); he proved that (1.2) has a solution in any box B with ∣B∣≥Cp2, for some constant C. The case k≥3 was a subject of investigation of a recent work of Ayyad and Cochrane [1]. They proved a number of results and conjectured that for fixed k≥3 and ε>0, if a,b,c∈Fp*, then there exists a solution of (1.1) in any box B with ∣B∣>Cp2+ε, for some C=C(ε,k). Given two sets A,B⊂Fp, the sum set A+B and the product set AB are defined as A+B={a+b;a∈A,b∈B},AB={ab;a∈A,b∈B}. For a given ξ∈Fp, we also use the notation ξA={ξa;a∈A}, so that the solvability of (1.1) can be restated in the form c∈a∏i=1kIi+b∏i=k+12kIi. In the present paper, we prove the following theorems which improve some results of Ayyad and Cochrane for k≥7 (see, [1, Table 1]). Theorem 1.1. For any ε>0, there exists δ=δ(ε)>0such that the following holds for any sufficiently large prime p: let I1,I2,…,I13⊂Fp*be intervals with ∣Ii∣>p1/4−δ,i=1,2,…,12;∣I13∣>p1/4+ε.Then for any a,b,c∈Fp*we have c∈a∏i=16Ii+b∏i=713Ii. If we allow 1∈I13, then the condition on the size of I13 can be relaxed to ∣I13∣>pε. Theorem 1.2. For any ε>0, there exists δ=δ(ε)>0such that the following holds for any sufficiently large prime p: let I1,I2,…,I13⊂Fp*be intervals with 1(modp)∈I13and ∣Ii∣>p1/4−δ,i=1,2,…,12;∣I13∣>pε.Then for any a,b,c∈Fp*, we have c∈a∏i=16Ii+b∏i=713Ii. From Theorem 1.2, we have the following consequence. Corollary 1.3. For any ε>0, there exists δ=δ(ε)>0such that the following holds for any sufficiently large prime p: let I1,I2,…,I12⊂Fp*be intervals satisfying 1(modp)∈I12and ∣Ii∣>p1/4−δ,i=1,2,…,11;∣I12∣>p1/4+ε.Then for any a,b,c∈Fp*, we have c∈a∏i=16Ii+b∏i=712Ii. Indeed, if we set I12′={1,2,…,⌊p1/4+ε/2⌋}(modp),I13={1,2,…,⌊pε/2⌋}(modp), then under the condition of Corollary 1.3 we have I12⊃I12′I13, and the claim follows from the application of Theorem 1.2. We remark that we state and prove our results for intervals of Fp* rather than of Fp just for the sake of simplicity. Indeed, this restriction is not essential, as any non-zero interval I⊂Fp contains an interval I′⊂I such that 0∉I′ and ∣I′∣≥∣I∣/3. In the case b=0, Eq. (1.1) describes the problem of representability of residue classes by product of variables from corresponding intervals. We shall consider the case when the variables are small positive integers. It is known from [8] that for any ε>0 and a sufficiently large cube-free m∈N, every λ with gcd(λ,m)=1 can be represented in the form x1…x8≡λ(modm),xi∈N,xi≤m1/4+ε. Under the same condition, Harman and Shparlinski [11] proved that λ can be represented in the form x1…x14≡λ(modm),xi∈N,xi≤m1/(4e1/2)+ε. We shall prove the following result. Theorem 1.4. For any 0<c0<1, there exists a positive integer n=n(c0)and a number δ=δ(c0)>0such that the following holds: let c0≤c<1and A={x(modm);1≤x≤mc,gcd(x,m)=1}.Then the set An is a subgroup of the multiplicative group Zm* and ∣An∣>δϕ(m). Here, as usual, ϕ(·) is Euler’s totient function, Zm* is the multiplicative group of invertible classes modulo m and An is the n-fold product set of A, that is, An={x1…xn;xi∈A}. Recall that ∣Zm*∣=ϕ(m). From Theorem 1.4, we shall derive the following consequences. Corollary 1.5. For any ε>0, there exists a positive integer k=k(ε)such that for any sufficiently large positive integer mthe congruence x1…xk≡1(modm),xi∈N,xi≤mεhas a solution with x1≠1. Corollary 1.6. There exists an absolute constant n0∈Nsuch that for any 0<ε<1and any sufficiently large prime p>p0(ε), every quadratic residue λmodulo pcan be represented in the form x1…xn0≡λ(modp),xi∈N,xi≤p1/(4e2/3)+ε. We remark that the constants n(c0),δ(c0) in Theorem 1.4, and k(ε) in Corollary 1.5 are effective and can be made explicit. 2. Proof of Theorems 1.1, 1.2 The proof of Theorems 1.1, 1.2 is based on the arguments of Ayyad and Cochrane [1] with some modifications. Lemma 2.1. Let N<pbe a positive integer, X⊂{1,2,…,p−1}. Then for any fixed integer constant n0>0, we have ∣{xy(modp);x∈X,1≤y≤N}∣>Δ∣X∣,where Δ=min{(p∣X∣)1/n0,N∣X∣1/n0}No(1)as N→∞. Proof We note that the statement is trivial for n0=1, so we can assume that n0>1. Let J be the number of solutions of the congruence x1y1≡x2y2(modp),x1,x2∈X,1≤y1,y2≤N. Then J=1p−1∑χ∣∑x∈Xχ(x)∣2∣∑y=1Nχ(y)∣2. Therefore, by the Hölder inequality, we get J≤A(n0−1)/n0B1/n0, (2.1) where A=1p−1∑χ∣∑x∈Xχ(x)∣2n0/(n0−1),B=1p−1∑χ∣∑y=1Nχ(y)∣2n0. (2.2) Next, we have A≤∣X∣2/(n0−1)(1p−1∑χ∣∑x∈Xχ(x)∣2)=∣X∣(n0+1)/(n0−1). The quantity B is equal to the number of solutions of the congruence y1…yn0≡yn0+1…y2n0(modp),1≤yi≤N. We express the congruence as the equation y1…yn0=yn0+1…y2n0+pz,1≤yi≤N,z∈Z. Note than ∣z∣≤Nn0/p. Hence, there are at most (2Nn0p+1)Nn0 possibilities for (yn0+1,…,y2n0,z). From the estimate for the divisor function it follows that, for each fixed yn0+1,…,y2n0,z, there are at most No(1) possibilities for y1,…,yn0. Therefore, B≤(Nn0p+1)Nn0+o(1). Incorporating this and (2.2) in (2.1), we obtain J≤∣X∣(n0+1)/n0(Np1/n0+1)N1+o(1). Therefore, from the relationship between the number of solutions of a symmetric congruence and the cardinality of the corresponding set, it follows ∣{xy(modp);x∈X,1≤y≤N}∣≥∣X∣2N2J≥min{∣X∣(n0−1)/n0p1/n0,∣X∣(n0−1)/n0N}No(1), which concludes the proof of Lemma 2.1.□ Lemma 2.2. Let X⊂{1,2,…,p−1}and let I⊂{1,2,…,p−1}be an interval with ∣I∣>p1/4+ε, where ε>0. Then ∣{xy(modp);x∈X,y∈I}∣>0.5min{p,∣X∣pc}for some c=c(ε)>0. Proof As in the proof of Lemma 2.1, we let J be the number of solutions of the congruence x1y1≡x2y2(modp),x1,x2∈X,y1,y2∈I. Then J=1p−1∑χ∣∑x∈Xχ(x)∣2∣∑y∈Iχ(y)∣2. Since ∣I∣>p1/4+ε, from the well-known character sum estimates of Burgess [5, 6], we have ∣∑n∈Iχ(n)∣<∣I∣p−δ,δ=δ(ε)>0, for any non-principal character χ(modp). Therefore, separating the term that corresponds to the principal character χ=χ0, we get J≤∣X∣2∣I∣2p−1+∣I∣2p−2δ(1p−1∑χ∣∑x∈Xχ(x)∣2)=∣X∣2∣I∣2p−1+∣X∣∣I∣2p−2δ. Hence, ∣{xy(modp);x∈X,y∈I}∣≥∣X∣2∣I∣2J≥0.5min{p,∣X∣pδ}. □ In what follows, the elements of Fp will be represented by their concrete representatives from the set of integers {0,1,…,p−1}. Following the lines of the work of Ayyad and Cochrane [1], we appeal to the result of Hart and Iosevich [12]. Lemma 2.3. Let A,B,C,Dbe subsets of Fp*satisfying ∣A∣∣B∣∣C∣∣D∣>p3.Then Fp*⊂AB+CD. We also need the following consequence of [4, Corollary 18]. Lemma 2.4. Let h<p1/4and let A1,A2,A3⊂Fp*be intervals of cardinalities ∣Ai∣>h,i=1,2,3. Then ∣A1A2A3∣≥exp(−Cloghloglogh)h3,for some constant C. Now we proceed to derive Theorems 1.1 and 1.2. Let p0.1<h<p1/4 to be defined later and assume that ∣Ii∣>h,i=1,2,…,12. Define X=I10I11I12,A=I1I2I3,B=I4I5I6,C=I7I8I9,D=XI13. From Lemma 2.4, we have that ∣X∣>h3+o(1) and ∣A∣∣B∣∣C∣>h9+o(1). Now we observe that Lemmas 2.1 and 2.2 imply that ∣D∣=∣XI13∣>h3+δ0 (2.3) for some δ0=δ0(ε)>0. Indeed, this is trivial for ∣X∣>h3.1, so let ∣X∣<h3.1. Then in the case of Theorem 1.1, the estimate (2.3) follows from Lemma 2.2. In the case of Theorem 1.2, we apply Lemma 2.1 with N=⌊pε⌋ and n0=⌈1/ε⌉, and obtain that ∣D∣>∣X∣∣I3∣δ>h3+0.9δ for some δ=δ(ε)>0. Thus, we have (2.3), whence ∣A∣∣B∣∣C∣∣D∣>h12+0.9δ0. Therefore, there exists c=c(ε)>0 such that if h=p14−c, then we get ∣A∣∣B∣∣C∣∣D∣>p3. Theorems 1.1 and 1.2 now follow by appealing to Lemma 2.3. 3. Proof of Theorem 1.4 Let G be a finite abelian group written multiplicatively and let X⊂G. The set X is a basis of order h for G if Xh=G. This definition implies that if 1∈X and X is a basis of order h for G, then X is also a basis of order h1 for G for any h1≥h. We need the following consequence of a result of Olson [13, Theorem 2.2] given in Hamidoune and Rödseth [10, Lemma 1]. Lemma 3.1. Let Xbe a subset of G. Suppose that 1∈Xand that Xgenerates G. Then Xis a basis for Gof order at most max{2,2∣G∣∣X∣−1}. We recall that Ψ(x;y) denotes the number of y-smooth positive integers n≤x (that is the number of positive integers n≤x with no prime divisors greater than y), and Ψq(x;y) denotes the number of y-smooth positive integers n≤x with gcd(n,q)=1. It is well known that for any ε>0, there exists δ=δ(ε)>0 such that Ψ(m;mε)≥δm. We need the following lemma, which follows from [7, Theorem 1]. Lemma 3.2. For any ε>0, there exists δ=δ(ε)>0such that Ψm(m;mε)>δϕ(m). We proceed to prove Theorem 1.4. Let S=S(c0,m) be the set of mc0-smooth positive integers n≤m with gcd(n,m)=1. As mentioned in [11], if x∈S, then we can combine the prime divisors of x in a greedy way into factors of size at most mc. More precisely, we can write x=x1…xk such that x1≤mc and mc/2≤xj≤mc for j=2,…,k. In particular, we have (k−1)c0/2≤(k−1)c/2≤1. Hence, k≤2/c0+1, and since 1(modm)∈A, it follows that S(modm)⊂An1;n1=⌈2/c0⌉+1. In particular, by Lemma 3.2, we have ∣An1∣≥∣S∣=Ψm(m;mc0)>δϕ(m) (3.1) for some δ=δ(c0)>0. Let h be the smallest positive integer such that An1h is a subgroup of Zm*. Applying Lemma 3.1 with G=An1h and X=An1, we get that h≤1+2∣An1h∣∣An1∣≤1+2∣Zm*∣∣An1∣<1+2ϕ(m)δϕ(m)=1+2δ−1. Therefore, since 1(modm)∈A, we get that for n=(1+⌈2δ−1⌉)n1, the set An is a multiplicative subgroup of Zm*. Taking into account (3.1), we conclude the proof of Theorem 1.4. Let now g be any element of the group An distinct from 1(modm). We also have that g−1∈An. Thus, Corollary 1.5, with k=2n, follows from the representation gg−1=1(modm). We shall now prove Corollary 1.6. Let A={x(modp);x∈N,x≤p1/(4e2/3)+ε}. In Theorem 1.4, we take m=p, c0=1/(4e2/3) and c=1/(4e2/3)+ε. Thus, there is an absolute constant n0 such that An0 is a subgroup of Fp* and ∣An0∣>δ0(p−1) for some absolute constant δ0>0. In other words, there is an integer ℓ∣p−1 with 1≤ℓ≤1/δ0 such that An0={xℓ(modp);1≤x≤p−1}. Let t=t(ℓ,p) be the smallest positive ℓ th power non-residue modulo p. According to the well-known consequence of Vinogradovʼs work [14] combined with the Burgess character sum estimate [5, 6], we have that t≤p1/(4e(ℓ−1)/ℓ)+ε/2. On the other hand, since t∉An0 we have t≥p1/(4e2/3)+ε. Hence, ℓ∈{1,2} and the claim follows. References 1 A. Ayyad and T. Cochrane , The congruence ax1…xk + bxk+1…x2k ≡ c (mod p) , Proc. Amer. Math. Soc. 145 ( 2017 ), 467 – 477 . Google Scholar Crossref Search ADS 2 A. Ayyad and T. Cochrane , Lattices in Z2 and the congruence xy + uv ≡ c (mod m) , Acta Arith . 132 ( 2008 ), 127 – 133 . Google Scholar Crossref Search ADS 3 A. Ayyad , T. Cochrane and Zh. Zheng , The congruence x1x2 ≡ x3x4 (mod p), the equation x1x2 = x3x4, and mean values of character sums , J. Number Theory 59 ( 1996 ), 398 – 413 . Google Scholar Crossref Search ADS 4 J. Bourgain , M. Z. Garaev , S. V. Konyagin and I. E. Shparlinski , On congruences with products of variables from short intervals and applications , Proc. Steklov Inst. Math. 280 ( 2013 ), 61 – 90 . Google Scholar Crossref Search ADS 5 D. A. Burgess , On character sums and primitive roots , Proc. London Math. Soc. 12 ( 1962 ), 179 – 192 . Google Scholar Crossref Search ADS 6 D. A. Burgess , On character sums and L-series. II , Proc. London Math. Soc. 13 ( 1963 ), 524 – 536 . Google Scholar Crossref Search ADS 7 E. Fouvry and G. Tenenbaum , Entiers sans grand facteur premier en progressions arithmetiques , Proc. London Math. Soc. (3) 63 ( 1991 ), 449 – 494 . Google Scholar Crossref Search ADS 8 M. Z. Garaev , On multiplicative congruences , Math. Z. 272 ( 2012 ), 473 – 482 . Google Scholar Crossref Search ADS 9 M. Z. Garaev and V. C. García , The equation x1x2 = x3x4 + λ in fields of prime order and applications , J. Number Theory 128 ( 2008 ), 2520 – 2537 . Google Scholar Crossref Search ADS 10 Y. O. Hamidoune and Ö. J. Rödseth , On bases for s-finite groups , Math. Scand. 78 ( 1996 ), 246 – 254 . Google Scholar Crossref Search ADS 11 G. Harman and I. E. Shparlinski , Products of small integers in residue classes and additive properties of Fermat quotients , Int. Math. Res. Not. 5 ( 2016 ), 1424 – 1446 . Google Scholar Crossref Search ADS 12 D. Hart and A. Iosevich , Sums and products in finite fields: an integral geometric viewpoint, Radon transforms, geometry, and wavelets, 129–135, Contemp. Math., 464, Amer. Math. Soc., Providence, RI, 2008 . 13 J. E. Olson , Sums of sets of group elements , Acta Arith. 28 ( 1975 /76), 147 – 156 . Google Scholar Crossref Search ADS 14 I. M. Vinogradov , On the bound of the least non-residue of n-th powers , Trans. Amer. Math. Soc. 29 ( 1927 ), 218 – 226 . © The Author(s) 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
The Quarterly Journal of Mathematics – Oxford University Press
Published: Sep 1, 2018
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