Concentration points on two and three dimensional modular hyperbolas and applications

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 $$xy\equiv \lambda (\mathrm{m}\mathrm{o}\mathrm{d}p),K+1\le x\le K+M,L+1\le y\le L+M$$ and for the number $I_3(M;L)$ of solutions of the congruence $$xyz\equiv \lambda (\mathrm{m}\mathrm{o}\mathrm{d}p),L+1\le x,y,z\le L+M\mathrm{.}$$ 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 $M<p^{1\mathrm{/}4},$ then $I_2(M;K,L)\le M^{o(1)}\mathrm{.}$ For $I_3(M;L)$ we prove that if $M<p^{1\mathrm{/}8}$ then $I_3(M;L)\le M^{o(1)}\mathrm{.}$ 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 ${\text{F}}_p^{\ast }$ of length $\mathrm{\mid }{\mathcal{I}}_i\mathrm{\mid }<p^{1\mathrm{/}8},$ then $$\mathrm{\mid }{\mathcal{I}}_1\cdot {\mathcal{I}}_2\cdot {\mathcal{I}}_3\mathrm{\mid }=(\mathrm{\mid }{\mathcal{I}}_1\mathrm{\mid }\cdot \mathrm{\mid }{\mathcal{I}}_2\mathrm{\mid }\cdot \mathrm{\mid }{\mathcal{I}}_3\mathrm{\mid }{)}^{1-o(1)}\mathrm{.}$$