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

Let p be a large prime number, K,L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M ; K, L) of solutions of the congruence xy ≡ λ (mod p), K + 1 ≤ x ≤ K + M, L + 1 ≤ y ≤ L + M and for the number I3(M ; L) of solutions of the congruence xyz ≡ λ (mod p), L + 1 ≤ x, y, z ≤ L + M. Using the idea of Heath-Brown from [6], we obtain a bound for I2(M ; K, L), which improves several recent results of Chan and Shparlinski [3]. For instance, we prove that if M < p1/4, then I2(M ; K, L) ≤ Mo(1). The problem with I3(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 < p1/8 one has I3(M ; L) ≤ Mo(1). Our results have applications to some other problems as well. For instance, it follows that if I1, I2, I3 are intervals in F∗p of length |Ii| < p1/8, then |I1 · I2 · I3| = (|I1| · |I2| · |I3|)1−o(1). MSC Classification: 11A07, 11B75