On Small Solutions to Quadratic Congruences

We estimate the deviation of the number of solutions of the congruence $$m^2-n^2 \equiv c \pmod q, \qquad 1 \le m \le M, \ 1\le n \le N,$$ 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 $\alpha$.