Concentration points on two and three dimensional modular hyperbolas and applications

TL;DR

This paper establishes sharp upper bounds for the number of solutions to certain modular hyperbola equations in two and three dimensions, improving previous results and with applications to product sets in finite fields.

Contribution

It provides new bounds for solution counts of modular hyperbolas, notably showing that for small intervals, the number of solutions is almost minimal, advancing understanding of modular hyperbola distributions.

Findings

For M < p^{1/4}, the number of solutions I_2(M;K,L) is essentially negligible.

For M < p^{1/8}, the number of solutions I_3(M;L) is essentially negligible.

Intervals of length less than p^{1/8} have product sets close to the expected size.

Abstract

Let $p$ be a large prime number, $K,L,M,\lambda$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(\lambda,p)=1.$ 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 $$ xy\equiv\lambda \pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M $$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equiv\lambda\pmod p, \quad L+1\le x,y,z\le L+M. $$ We obtain a bound for $I_2(M;K,L),$ which improves several recent results of Chan and Shparlinski. For instance, we prove that if $M<p^{1/4},$ then $I_2(M;K,L)\le M^{o(1)}.$ For $I_3(M;L)$ we prove that if $M<p^{1/8}$ then $I_3(M;L)\le M^{o(1)}.$ Our results have applications to some other problems as well. For instance, it follows that if $\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3$ are intervals in $\F^*_p$ of length $|\mathcal{I}_i|< p^{1/8},$ then $$ |\mathcal{I}_1\cdot \mathcal{I}_2\cdot \mathcal{I}_3|= (|\mathcal{I}_1|\cdot |\mathcal{I}_2|\cdot |\mathcal{I}_3|)^{1-o(1)}. $$