Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers

TL;DR

This paper investigates the asymptotic distribution of small solutions to quadratic congruences in three variables modulo prime powers, establishing new bounds for solution existence depending on fixed or varying coefficients.

Contribution

It provides asymptotic formulas for solutions of quadratic congruences modulo prime powers, extending previous results by relaxing bounds on the size of solutions.

Findings

Asymptotic formula established for fixed coefficients with N ≫ q^{1/2+ε}.

Extended results to variable coefficients with N ≫ q^{11/18+ε}.

Compared bounds with Heath-Brown's results for square-free moduli.

Abstract

Let $p>5$ be a fixed prime and assume that $\alpha_1,\alpha_2,\alpha_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $\alpha_1,\alpha_2,\alpha_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\varepsilon}$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$.