Diophantine approximation with sums of two squares II

Stephan Baier; Habibur Rahaman

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arXiv · 2508.18044

Diophantine approximation with sums of two squares II

Stephan Baier·Habibur Rahaman

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Abstract

Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n|| 0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $\gamma<3/7-\varepsilon$. In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$, where we confine ourselves to the particular case of sums of two squares.

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Stephan Baier, Habibur Rahaman. 2025-08-25. Diophantine approximation with sums of two squares II. https://arxiv.org/abs/2508.18044

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