Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime

Transcription

1 Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime J. Cilleruelo and M. Z. Garaev Abstract We obtain a sharp upper bound estimate of the form Hp o(1) for the number of solutions of the congruence x yr (mod p); x, y N, x, y H, r U for certain ranges of H and U, where U is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use this bound to show that almost all the residue classes modulo p can be represented in the form xg y (mod p) with positive integers x < p 5/8+ε and y < p 3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzg t (mod p) with positive integers x, y, z, t < p 1/4+ε. In particular any λ 0 (mod p) can be represented in the form λ xyzuvwg t (mod p) for some positive integers x, y, z, t, u, v, w < p 1/4+ε. 1 Introduction In what follows, ε is a small fixed positive quantity, F p is the field of residue classes modulo a prime number p, which we consider to be sufficiently large in terms of ε. The distributional properties of powers of a primitive root modulo p and subgroups of F p has a long story, starting from the work of Vinogradov [13] in A substantial amount of information and results can be can be found 1

2 in the book [11]. In the present paper we continue the investigation on this topic. Our first result is closely related to the work of Bourgain, Konyagin and Shparlisnki [2]. Theorem 1. Let U F p and n, H be positive integers such that U < p n/(2n+1) ; U U < 10 U ; U H n < p. Then the number J of solutions of the congruence satisfies J H. x yr (mod p); x, y N, x, y H, r U, (1) Here and below, the notaion A B is used to denote that A < B p o(1), or equivalently, for any ε > 0 there is a constant c = c(ε) such that A < c B p ε. We also recall that given sets A and B their product-set A B is defined by A B = {ab; a A, b B}. The proof of Theorem 1 is based on ideas and results of Bourgain, Konyagin and Shparlinski [2]. Nevertheless, in the indicated range our Theorem 1 in certain sense is optimal and improves one of the main results of [2]. We give several applications of Theorem 1. Let I = {1, 2, 3,..., H} (mod p) be an interval of F p with I = H elements. Denote by G either a subgroup of F p or the set {1, g, g 2,..., g N 1 } (mod p) with G = N elements, formed with powers of a primitive root g modulo p. Theorem 2. For any fixed ε > 0, if I > p 5/8+ε, G > p 3/8, then for some δ = δ(ε) > 0. I G = p + O(p 1 δ ) Theorem 3. For any fixed ε > 0, if I > p 1/4+ε, G > p 1/4, then for some δ = δ(ε) > 0. I I I G = p + O(p 1 δ ) 2

3 Let us mention several results relevant to Theorems 2, 3. In [9] it was shown that if I > p ε and A is an arbitrary subset of F p with A > p ε then I A = p + O(p 1 δ ); δ = δ(ε) > 0. Later, the exponent was improved by Bourgain to 5 8 (unpublished). We also mention that if I > p 1/2+ε, then one has I I = p + O(p 1 δ ); δ = δ(ε) > 0, see, for example, [10]. Theorem 3 can be compared with the result from [8], where it was shown that under the same condition I > p 1/4+ε one has I I I I = p + O(p 1 δ ); δ = δ(ε) > 0. However, the presence of G in our theorems is an additional obstacle which we are able to overcome using Theorem 1. Theorem 3 implies, in particular, that any λ 0 (mod p) can be represented in the form λ xyzuvwg t (mod p) for some positive integers x, y, z, t, u, v, w < p 1/4+ε. In passing, we remark that in Theorem 1 the condition U U < 10 U can be relaxed up to U U < U p o(1). However, the formulation in the form U U < 10 U already applies for our set G and is sufficient to prove Theorems 2, 3. 2 Auxiliary Lemmas We start with the following lemma of Bourgain, Konyagin and Shparlinski [2], see also [5] for a different proof with refined constants. Lemma 1. Let { r } A = s ; r, s N, gcd(r, s) = 1, r, s Q be the set of Farey fractions of order Q and m be a given positive integer. Then the m-fold product set A (m) of A satisfies ( A (m) log Q > exp C(m) ) A m log log Q where C(m) > 0 depends only on m, provided that Q is large enough. 3

4 We recall that the m-fold product set A (m) of A is defined as A (m) = {a 1 a m ; a 1,..., a m A}. Our next lemma stems from the work of Bourgain et. al. [3]. Recall that a lattice in R n is an additive subgroup of R n generated by n linearly independent vectors. Take an arbitrary convex compact and symmetric with respect to 0 body D R n. Recall that, for a lattice Γ R n and i = 1,..., n, the i-th successive minimum λ i (D, Γ) of the set D with respect to the lattice Γ is defined as the minimal number λ such that the set λd contains i linearly independent vectors of the lattice Γ. Obviously, λ 1 (D, Γ)... λ n (D, Γ). From [1, Proposition 2.1] it is known that n min{λ i (D, Γ), 1} i=1 (2n + 1)!!. (2) D Γ Lemma 2. For any s 0 F p, the number of solutions of the congruence x s 0 y (mod p); x, y N, x X, y Y, gcd(x, y) = 1, (3) is bounded by O(1 + XY p 1 ). Proof. Consider the lattice and the body Γ = {(u, v) Z 2 ; u s 0 v (mod p)} D = {(u, v) R 2 ; u X, v Y }. Let λ 1, λ 2 be the consecutive minimas of the body D with respect to the lattice Γ. If λ 2 > 1 then there is at most one independent vector in Γ D, implying that J 1, where J is the number of solutions of (3). Let now λ 2 1. Then, by (2), we get λ 1 λ 2 15 Γ D 15 J. Let (u i, v i ) λ i D Γ, i = 1, 2, be linearly independent. Then ( ) u1 v 0 det 1 0 (mod p). u 2 v 2 4

5 Therefore, p ( u1 v 1 det u 2 v 2 and the result follows. ) = u1 v 2 u 2 v 1 2λ 1 λ 2 XY 30XY J We also need the following simple lemma. Lemma 3. Let X, Y be positive numbers with XY < p. Then for any λ there is at most one solution to the congruence x y λ (mod p); x, y N, x X, y Y, gcd(x, y) = 1. Proof. Assuming that there is at least one solution (x 0, y 0 ), we get that xy 0 x 0 y (mod p), and since the both hand sides are not greater than XY < p, the congruence is converted to an equality, which together with gcd(x, y) = gcd(x 0, y 0 ) = 1 implies that x = x 0, y = y 0. 3 Proof of Theorem 1 Given a positive integer d we let J d (H, U) be the number of solutions of (1) with the additional condition gcd(x, y) = d. Then we have J = d H J d (H, U) = d H J 1 (H/d, U). (4) Thus, it suffices to prove that for any δ > 0 there exists c = c(δ) > 0 such that J 1 (H/d, U) < (H/d)p δ. In particular, since J 1 (H/d, U) (H/d) 2, we can assume that H/d > p δ. Let m be the smallest positive integer such that U < (H/d) m. Clearly, m < 1 + 1/δ. It is easy to see that (H/d) m < p/ U. Indeed, if (H/d) m p/ U, then from the condition of the theorem it follows that m n + 1. On the other hand, by the definition of m we have (H/d) m 1 U. Hence, the lower and the upper bounds for H/d give U p (m 1)/(2m 1) p n/(2n+1), 5

6 which contradicts the condition of the theorem. Thus, we have U < (H/d) m < p/ U. (5) Since each pair of relatively prime positive integers (x, y) can be defined by the rational number x/y, it follows that J 1 (H/d, U) is equal to the cardinality of the set { x J d = y ; x, y N, x, y H d, gcd(x, y) = 1, x } y (mod p) U. We observe that the m-fold product set J (m) d satisfies { u v ; u, v N, u, v (H/d)m, gcd(u, v) = 1, u } v (mod p) U (m). J (m) d The Plünecke inequality (see, [12, Theorem 7.7]) together with the condition U U < 10 U implies that U (m) < 10 m U. Thus, using Lemma 2 and the inequality (5), we derive that J (m) d r U (m) ) ) (1 + (H/d)2m U (1 + (H/d)2m (H/d) m. p p On the other hand Lemma 1 implies that J (m) d J d m. Thus, J 1 (H/d, U) = J d H d, which, in view of (4), finishes the proof of our theorem. 4 Preliminary consequences of Theorem 1 From Theorem 1 we get the following consequence. Corollary 1. Let U F p and n, H be positive integers with U < p n/(2n+1), U U 10 U, U H n < p. Then the number J of solutions of the congruence satisfies J H U. xr x 1 r 1 (mod p); 1 x, x 1 H, r, r 1 U (6) 6

7 Indeed, we can fix r = r 0 U such that J U J, where J is the number of solutions of the congruence Now we simply denote x x 1 r 1 0 r 1 (mod p); 1 x, x 1 H, r 1 U. U = {r 1 0 r 1 ; r 1 U} and apply Theorem 1 with U substituted by U. Using Corollary 1, we shall establish the following statement, which will be used to prove Theorem 2. Lemma 4. Let 0 < ε < 0.01 be fixed, U F p be such that and H = p 1/4+ε, 2H < U p 3/8 0.5ε. Then the number T of solutions of the congruence U U 10 U qxr q 1 x 1 r 1 (mod p) (7) in positive integers x, x 1, prime numbers q, q 1 and elements r, r 1 U with satisfies x, x 1 H, U 2 < q, q 1 U (8) T U 2 H. Proof. We have T = T 1 + T 2, where T 1 is the number of solutions of (7) satisfying (8) with the additional condition q = q 1, and T 2 is the number of solutions with q q 1. We observe that U < p 3/8 < p 2/5 and U H 2 < p 3/8 0.5ɛ p 1/2+2ɛ < p 7/8+3ɛ/2 < p. Thus, we can apply Corollary 1 with n = 2 and get T 1 U 2 H. 7

8 In order to estimate T 2, we fix x 1, r, r 1 such that T 2 U 2 HT 2, where T 2 is the number of solutions of the congruence qx x 1r 1 q 1 r (mod p) in positive integers x H and prime numbers q, q 1 with q q 1, 0.5 U < q, q 1 U. From x H < q 1, it follows that gcd(qx, q 1 ) = 1. Since H U 2 < p, from Lemma 3 we get that xq and q 1 are uniquely determined. Since x < q, the value xq uniquely determines x and q. Hence, T 2 1, whence T 2 U 2 H concluding the proof of our lemma. Lemma 5. Let 0 < ε < 0.01 be fixed, U F p be such that and H = p 1/4+ε, Q = p 1/4, U p 1/4 ε. Then the number T of solutions of the congruence U U 10 U q 1 q 2 xr q 1q 2x r (mod p) (9) in positive integers x, x, prime numbers q 1, q 2, q 1, q 2 and elements r, r U with x, x H, Q 4 < q 1, q 1 < Q 2, Q 2 < q 2, q 2 < Q, (10) satisfies T Q 2 H U. We have T = T 1 + T 2 + T 3 + T 4, (11) where T 1 is the number of solutions of (9) satisfying (10) and with the additional condition q 1 = q 2, q 1 = q 2, T 2 is the number of solutions with q 1 q 1, q 2 q 2, T 3 is the number of solutions with q 1 = q 1, q 2 q 2 and T 4 is the number of solutions with q 1 q 1, q 2 = q 2. 8

9 We have T 1 Q 2 T 1, where T 1 is the number of solutions of the congruence xr x r (mod p); x, x H, r, r U. Applying Corollary 1 with n = 1 or n = 2, we get that T 1 U H. Therefore, T 1 Q 2 H U. (12) To estimate T 2, we fix x, r, x, r and see that T 2 U 2 H 2 T 2, where T 2 is the number of solutions of the congruence q 1 q 2 q 1q 2 x r xr (mod p) in prime numbers q 1, q 1, q 2, q 2 with Q 4 < q 1, q 1 < Q 2, Q 2 < q 2, q 2 < Q, q 1 q 1, q 2 q 2. Since gcd(q 1 q 2, q 1q 2) = 1 and Q 4 < p, it follows from Lemma 3 that the numbers q 1 q 2 and q 1q 2 are uniquely determined. Since q 1 < q 2 and q 1 < q 2, this implies that in fact all the prime numbers q 1, q 2, q 1, q 2 are uniquely determined. Therefore, T 2 1 and we get T 2 U 2 H 2 Q 2 H U. (13) In order to estimate T 3 and T 4, we note that T 3 + T 4 QT 5, where T 5 is the number of solutions of the congruence qxr q x r (mod p), in positive integers x, x, prime numbers q, q and elements r, r U with x, x H, Q 4 < q, q < Q, q q. 9

10 We introduce variables x 1, x 2 with and note that x 1 = qx, x 2 = q x x 1 r x 2 r (mod p); x 1, x 2 QH, r, r U. We can apply Corollary 1 with n = 1 and H substituted by QH (clearly, the conditions of Corollary 1 are satisfied). It then follows that there are at most QH U p o(1) possibilities for the quadruple (x 1, x 2, r, r ). Each such quadruple determines q, x, q, x with at most p o(1) possibilities, because q, x, q, x are divisors of x 1 x 2 < p. Therefore, we get that implying that T 5 QH U, T 3 + T 4 Q 2 H U. Inserting this estimate together with (12) and (13) into (11), we conclude the proof of our lemma. 5 Proof of Theorem 2 Assuming ε < 0.01, we define m 0 = 1 ε, L = p 1/m 0, H = p 1/4+ε, N = p 3/8 0.5ε. From the well-known character sum estimate of Burgess [4] it is known that there exists δ = δ(ε) > 0 such that for any non-principal character χ modulo p the following bound holds: χ(x) H1 δ. (14) x H Let G be a subset of G such that G = N and G G 2 G. The existence of such a subset is obvious, since either G consists on consecutive powers of a primitive root or it is a subgroup of F p, which is cyclic and therefore consists of all powers of some element. 10

11 It suffices to prove that for some δ 0 = δ 0 (ε) > 0 there are p + O(p 1 δ 0 ) residue classes modulo p of the form zxqr (mod p), with positive integers x, z, prime numbers q and elements r satisfying z L, x H, N 2 < q < N, r G. Let Λ F p be the exceptional set, that is, assume that the congruence zxqr λ (mod p) has no solutions in λ Λ and z, x, q, r as above. We write this condition in the form of character sums 1 χ(zxqrλ 1 ) = 0, p 1 χ z L x H 0.5N<q<N q is prime r G λ Λ where χ runs through the set of characters modulo p. Separating the term corresponding to the principal character χ = χ 0 and recalling that there are more than 0.1N/ log N prime numbers q with 0.5N < q < N, we get LHN 2 Λ χ(z) χ(λ). χ χ 0 z L x H 0.5N<q<N q is prime r G χ(xqr) Following [6, 7], we split the set of nonprincipal characters χ into two subsets X 1 and X 2 as follows. To the set X 1 we allot those characters χ, for which χ(z) L 1 0.1δ, z L where δ is defined from (14). The remaining characters we include to the set X 2, these are the characters that satisfy χ(z) < L 1 0.1δ. z L λ Λ Thus, we have where W i = χ(z) χ X i z L x H LHN 2 Λ W 1 + W 2, (15) 0.5N<q<N q is prime 11 χ(xqr) χ(λ). r G λ Λ

12 To deal with W 1, we show that the cardinality of X 1 is small. We have X 1 L 2m0(1 0.1δ) χ(z) 2m 0 χ(z) 2m 0 = (p 1)T, χ X 1 z L χ z L where T is the number of solutions of the congruence x 1 x m0 y 1 y m0 (mod p); x i, y j N; x i, y j L. Since L m 0 < p, the congruence is converted to an equality and we have, by the estimate for the divisor function, at most L m0+o(1) solutions. Thus, X 1 L 2m 0(1 0.1δ) pl m 0+o(1), whence, in view of L m 0 = p 1+o(1), we get X 1 p 0.2δ. Thus, estimating in W 1 the sums over z, q, r, λ trivially and applying (14) to the sum over x, we get W 1 = χ(z) χ(x) χ(qr) χ(λ) χ X 1 z L x H 0.5N<q<N r G λ Λ q is prime X 1 L N 2 Λ max χ(x) p 0.2δ LN 2 Λ H 1 δ. χ χ 0 x H Therefore, since H > p 1/4 we have, for sufficiently large p, the estimate Inserting this bound into (15), we get W 1 < LHN 2 Λ p 0.01δ. LHN 2 Λ W 2. (16) We next estimate W 2. By the definition of X 2, we have W 2 L 1 0.1δ χ(xqr) χ(λ). (17) χ x H r G λ Λ 0.5N<q<N q is prime 12

13 Next, we have and χ x H χ(λ) 2 = (p 1) Λ λ Λ χ 0.5N<q<N q is prime χ(xqr) 2 = (p 1)T, r G where T is the number of solutions of the congruence xqr x 1 q 1 r 1 (mod p), in positive integers x, x 1 prime numbers q, q 1 and elements r, r 1 G satisfying x 1, x 2 H, 0.5N < q, q 1 < N, r, r 1 G. From Lemma 4 with U = G it follows that T N 2 H. Therefore, applying the Cauchy-Schwarz inequality in (17), and using (16), we obtain that Thus, Therefore, L 2 H 2 N 4 Λ 2 W 2 2 L δ (p 1) 2 Λ T L δ p 2 Λ N 2 H. Λ p2 L 0.02δ HN 2 pl 0.02δ. Λ < p 1 δ 0 for some δ 0 = δ 0 (ɛ), which concludes the proof. 6 Proof of Theorem 3 The proof follows the same line as the proof of Theorem 2, using Lemma 5 instead of Lemma 4. Assuming ε < 0.01, we define m 0 = 1 ε, L = p 1/m 0, H = p 1/4+ε, Q = p 1/4, N = p 1/4 ε. 13

14 Following the proof of Theorem 2, we denote by δ = δ(ε) > 0 a positive constant such that for any non-principal character χ modulo p the following bound holds: χ(x) H1 δ. (18) x H We again let G be a subset of G such that G = N and G G 2 G. It suffices to prove that for some δ 0 = δ 0 (ε) > 0 there are p + O(p 1 δ 0 ) residue classes modulo p of the form zxq 1 q 2 r (mod p), with positive integers z, x, prime numbers q 1, q 2 and elements r satisfying z L, x H, Q 4 < q 1 < Q 2, Q 2 < q 2 < Q, r G. Let Λ F p be the exceptional set, that is, assume that the congruence zxq 1 q 2 r λ (mod p) has no solutions in λ Λ and z, x, q 1, q 2, r as above. Following the proof of Theorem 2, we derive that LHQ 2 N Λ χ(z) χ(xq 1 q 2 r) χ(λ). χ χ 0 z L x H r G λ Λ Q/4<q 1 <Q/2 Q/2<q 2 <Q q 1,q 2 are primes We define the set of characters X 1 and X 2 exactly the same way as in the proof of Theorem 3 and then write LHQ 2 N Λ W 1 + W 2, (19) where W i = χ(z) χ X i z L x H Q/4<q 1 <Q/2 Q/2<q 2 <Q q 1,q 2 are primes χ(xq 1 q 2 r) χ(λ). r G λ Λ From the proof of Theorem 2 it follows that X 1 p 0.2δ. 14

15 Thus, estimating in W 1 the sums over z, q 1, q 2, r, λ trivially and applying (18) to the sum over x, we get Inserting this bound into (19), we get W 1 < LHQ 2 N Λ p 0.01δ. LHQ 2 N Λ W 2. Following the argument of the proof of Theorem 2 we have L 2 H 2 Q 4 N 2 Λ 2 W 2 2 L δ (p 1) 2 Λ T, where T is the number of solutions of the congruence xq 1 q 2 r x q 1q 2r (mod p), in positive integers x, x prime numbers q 1, q 2, q 1, q 2 and elements r, r 1 G satisfying x 1, x 2 H, Q 4 < q 1, q 1 < Q 2, Q 2 < q 2, q 2 < Q, r, r 1 G. From Lemma 5 applied with U = G it follows that Therefore, Thus, whence T HQ 2 N. L 2 H 2 Q 4 N 2 Λ 2 L δ p 2 HQ 2 Λ. Λ p2 HQ 2 N L 0.02δ, Λ < p 1 δ 0 for some δ 0 = δ 0 (ɛ). This finishes the proof of Theorem 3. 15

16 References [1] U. Betke, M. Henk and J. M. Wills, Successive-minima-type inequalities, Discr. Comput. Geom., 9 (1993), [2] J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm, Int. Math. Res. Not. (2008), Art. ID rnn 090, 29 pp. [3] 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), [4] D. A. Burgess, On character sums and L-series, Proc. London Math. Soc., 12 (1962), [5] J. Cilleruelo, A note on product sets of fractions, Preprint. [6] J. Cilleruelo and M. Z. Garaev, Least totients in arithmetic progressions, Proc. Amer. Math. Soc., 137 (2009), [7] M. Z. Garaev, A note on the least totient of a residue class, Quart. J. Math., 60 (2009), [8] M. Z. Garaev, On multiplicative congruences, Math. Zeitschrift., 272 (2012), [9] M. Z. Garaev and A. A. Karatsuba, On character sums and the exceptional set of a congruence problema, J. Number Theory, 114 (2005), [10] M. Z. Garaev and A. A. Karatsuba, The representation of residue classes by products of small integers, Proc. Edinburgh Math. Soc., 50 (2007), [11] S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Univ. Press, Cambridge,

17 [12] M. B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets, Grad. Texts in Math., Vol 165, Springer-Verlag, New York 1996, 293 pp. [13] I. M. Vinogradov, On the distribution of indices, Doklady Akad. Nauk SSSR, 20 (1926), (in Russian). Address of the authors: J. Cilleruelo, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid , Spain. M. Z. Garaev, Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, C.P , Morelia, Michoacán, México, 17