Diophantine approximation with sums of two squares II

Abstract

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

This is a reworked version. We have managed to establish the same result by simpler means, resulting in a reduction of pages from 17 to 15