Diophantine approximation with sums of two squares

Stephan Baier; Habibur Rahaman

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arXiv · 2504.09650

Diophantine approximation with sums of two squares

Stephan Baier·Habibur Rahaman

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Abstract

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||\alpha n|| 0$. This is an easy consequence of a result by Cook on small fractional parts of diagonal quadratic forms. Secondly, we give a quantitative version with a lower bound of this result when the exponent $1/2-\varepsilon$ is replaced by any fixed $\gamma<3/7$. To this end, we use the Voronoi summation formula and a bound for bilinear forms with Kloosterman sums to fixed moduli by Kerr, Shparlinski, Wu and Xi.

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Stephan Baier, Habibur Rahaman. 2025-04-13. Diophantine approximation with sums of two squares. https://arxiv.org/abs/2504.09650

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