Small solutions of ternary quadratic congruences with averaging over the moduli
Stephan Baier, Aishik Chattopadhyay
TL;DR
This work advances small-solution results for ternary quadratic congruences by averaging over moduli. By introducing a dyadic averaging in $q$ and analyzing weighted counts with smoothed cutoffs, the authors reduce the problem to bounding a discrepancy term via Dirichlet characters and, crucially, the eighth moment of Dirichlet $L$-functions. The core novelty is an eighth-moment bound applied to short character sums across moduli in a dyadic range, enabling a new height bound of $Q^{3/8+\varepsilon}(\alpha_2 Q)^{\varepsilon}$ for almost all pairs $(q,\alpha_3)$ with $Q<s\le 2Q$ and $(q,2\alpha_2)=1$. This unconditional improvement leverages Burgess-type estimates, Mellin transform techniques, and a delicate combination of smoothed and sharp-cutoff analyses to establish asymptotics for the weighted counts and to break the previous barrier $11/24$ in the exponent when averaging moduli.
Abstract
In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed $α_2\in \mathbb{N}$ 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 $\max\{|x_1|,|x_2|,|x_3|\}\le q^{11/24+\varepsilon}$ for, in a sense, almost all $α_3$, where $α_3$ 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 $\le Q^{3/8+\varepsilon}α_2^{\varepsilon}$ for almost all pairs $(q,α_3)$, with $Q$ large enough, $Q<q\le 2Q$, $q$ coprime to $2α_2$ and $α_3$ running over the reduced residue classes modulo $q$.