Diophantine approximation with sums of two squares II

TL;DR

This paper advances the understanding of Diophantine approximation by providing a quantitative result for sums of two squares, showing how well irrational numbers can be approximated by such sums with a specific error bound.

Contribution

It establishes a quantitative approximation result for sums of two squares, refining previous bounds specifically for this quadratic form.

Findings

Proves a quantitative approximation bound for sums of two squares.

Shows the approximation rate can be achieved with an error less than n^{-(1/2- ext{epsilon})}.

Extends previous results to a specific quadratic form case.

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 $||\alpha n||<n^{-(1/2-\varepsilon)}$ for any fixed $\varepsilon>0$. 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.