Type: Article

Concentration of points on two and three dimensional modular hyperbolas and applications

Javier Cilleruelo; Moubariz Z. Garaev;

Abstract:

Let p be a large prime number, K, L, M, lambda be integers with 1 <= M <= p and gcd(lambda, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I(2)(M; K, L) of solutions of the congruence xy equivalent to lambda (mod p), K + 1 <= x <= K + M, L + 1 <= y <= L + M, and for the number I(3)(M; L) of solutions of the congruence xyz equivalent to lambda (mod p), L + 1 <= x, y, z <= L + M. Using the idea of Heath-Brown from [H], we obtain a bound for I(2)(M; K, L), which improves several recent results of Chan and Shparlinski [CS]. For instance, we prove that if M < p(1/4), then I(2)(M; K, L) <= M(o(1)). The problem with I(3)(M; L) is more difficult and requires a different approach. Here, we connect this problem with the Pell diophantine equation and prove that for M < p(1/8) one has I(3)(M; L) <= M(o(1)). Our results have applications to some other problems as well. For instance, it follows that if I(1), I(2), I(3) are intervals in F(p)* of length vertical bar I(i)vertical bar < p(1/8), then vertical bar I(1) . I(2) . I(3)vertical bar = (vertical bar I(1)vertical bar . vertical bar I(2)vertical bar . vertical bar I(3)vertical bar)(1-o(1)).