On Small Solutions to Quadratic Congruences

TL;DR

This paper investigates the distribution of solutions to quadratic congruences and their deviation from expected values, connecting these findings to the pair correlation problem of fractional parts of quadratic functions.

Contribution

It provides new estimates for the deviation of solution counts of quadratic congruences, linking them to pair correlation problems for quadratic fractional parts.

Findings

Estimated the deviation of solution counts from the expected value.

Connected solution distribution to pair correlation of quadratic fractional parts.

Motivated by Heath-Brown's recent results on quadratic distributions.

Abstract

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 $\alpha$.