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 $\mathrm{\mid }\mathrm{\mid }\alpha n\mathrm{\mid }\mathrm{\mid }<n^{-(1\mathrm{/}2-\epsilon )}$ for any fixed $\epsilon >0$. We also provided a quantitative version with a lower bound when the exponent $1\mathrm{/}2-\epsilon$ is replaced by a smaller exponent $\gamma <3\mathrm{/}7-\epsilon$. In this article, we establish a quantitative version for the exponent $1\mathrm{/}2-\epsilon$, where we confine ourselves to the particular case of sums of two squares.