Diophantine approximation with sums of two squares
Stephan Baier, Habibur Rahaman
TL;DR
The paper investigates Diophantine approximation for integers $n$ represented by a positive definite integral binary quadratic form $Q$. It proves two main results: (i) for any irrational $\alpha$, there are infinitely many $n$ in $\mathcal{A}_Q$ with $||\alpha n||<n^{-1/2+\varepsilon}$, derived from Cook's bounds on diagonal forms; (ii) a quantitative version with a lower bound for the count of such $n$ when the exponent is any fixed $\gamma<3/7$, obtained by combining the Voronoi summation formula for $Q$ with fixed-modulus bilinear estimates for Kloosterman sums due to Kerr–Shparlinski–Wu–Xi. The authors reduce the problem to an asymptotic for a sum counting representations of integers in unions of residue classes, then carefully analyze the main term and an intricate error term using Gauss and Kloosterman-type sums, Bessel transforms, and Dirichlet-approximation techniques. This yields an explicit lower bound and, in particular, improves upon prior $1/3$-type exponents by achieving $3/7$ in the bounded-exponent regime, with additional information about the density of suitable $n$.
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 $α$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||αn||<n^{-(1/2-\varepsilon)}$ 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 $γ<3/7$. 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.