Diophantine approximation with sums of two squares II

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.