Diophantine approximation with integers represented by binary quadratic forms

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