Small solutions of ternary quadratic congruences with averaging over the moduli
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arXiv
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2025
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| author | Baier, Stephan Chattopadhyay, Aishik |
| author_facet | Baier, Stephan Chattopadhyay, Aishik |
| contents | 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 $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small solutions of ternary quadratic congruences with averaging over the moduli Baier, Stephan Chattopadhyay, Aishik Number Theory 11D79, 11E04, 11E25, 11L40, 11M06, 11T24 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 $q$. |
| title | Small solutions of ternary quadratic congruences with averaging over the moduli |
| topic | Number Theory 11D79, 11E04, 11E25, 11L40, 11M06, 11T24 |
| url | https://arxiv.org/abs/2509.16980 |