Diophantine approximation with sums of two squares

Abstract

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 $\hat{I}\pm$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||\hat{I}\pm n||<n^{-(1/2-\epsilon )}$ for any fixed but arbitrarily small $\epsilon >0$. 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 $1/2-\epsilon$ is replaced by any fixed ${\hat{I}}^3<3/7$. 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.

17 pages, major revisions