Small solutions of ternary quadratic congruences with averaging over the moduli

Abstract

In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed $\hat{I}{\pm }_2\in \mathbb{N}$ coprime to $q$, the congruence \[ x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q} \] has a solution of $(x_1,x_2,x_3)\in {\mathbb{Z}}^3$ with $x_3$ coprime to $q$ of height $\max \{|x_1|,|x_2|,|x_3|\}\le q^{11/24+\epsilon }$ for, in a sense, almost all $\hat{I}{\pm }_3$, where $\hat{I}{\pm }_3$ runs over the reduced residue classes modulo $q$. Here it was of significance that $11/24<1/2$, so we broke a natural barrier. In this paper, we average the moduli $q$ in addition, establishing the existence of a solution of height $\le Q^{3/8+\epsilon }\hat{I}{\pm }_2^{\epsilon }$ for almost all pairs $(q,\hat{I}{\pm }_3)$, with $Q$ large enough, $Q<q\le 2Q$, $q$ coprime to $2\hat{I}{\pm }_2$ and $\hat{I}{\pm }_3$ running over the reduced residue classes modulo $q$.

11 pages