Concentration points on two and three dimensional modular hyperbolas and applications

Let $p$ be a large prime number, $K,L,M,λ$ be integers with $1\le M\le p$ and ${\color{red}\gcd}(λ,p)=1.$ 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 $xy\equivλ\pmod p, \qquad K+1\le x\le K+M,\quad L+1\le y\le L+M$ and for the number $I_3(M;L)$ of solutions of the congruence $xyz\equivλ\pmod p, \quad L+1\le x,y,z\le L+M.$ 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 $|\mathcal{I}_1\cdot \mathcal{I}_2\cdot \mathcal{I}_3|= (|\mathcal{I}_1|\cdot |\mathcal{I}_2|\cdot |\mathcal{I}_3|)^{1-o(1)}.$

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