Small solutions of generic ternary quadratic congruences to general moduli

TL;DR

This paper investigates the distribution of small solutions to certain quadratic congruences modulo large odd integers, extending previous prime power results and breaking the 1/2 barrier in the solution size exponent.

Contribution

The authors develop new asymptotic formulas for solutions to quadratic congruences with general moduli, utilizing advanced character sum estimates and breaking previous size barriers.

Findings

Established asymptotic formulas for solutions with size N ≥ q^{11/24+ε}

Extended previous prime power results to general moduli q

Broken the 1/2 barrier in the solution size exponent

Abstract

We study small non-trivial solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli $q$. Above, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume that $(\alpha_2\alpha_3,q)=1$. We show that for all $\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients.