Concentration points on two and three dimensional modular hyperbolas and applications

2010/07/09 by Javier Cilleruelo, Cilleruelo, J., M. Z. Garaev +1 · 1 citation
Mathematics · #11A07 #11B75 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1007.1526

openalex publication_date 2010/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a large prime number, K,L,M,λ be integers with 1≤ M≤ p and \colorredgcd(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\pmod 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\pmod p, L+1≤ x,y,z≤ L+M. We obtain a bound for I₂(M;K,L), which improves several recent results of Chan and Shparlinski. For instance, we prove that if M

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