On Small Solutions to Quadratic Congruences

Abstract

We estimate the deviation of the number of solutions of the congruence $$m^2-n^2\equiv c(\mathrm{mod}q),1\le m\le M,1\le n\le N,$$ from its expected value on average over $c=1,...,q$. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function $\alpha k^2$, $k=1,2,...$ with a real $\alpha$.