Diophantine approximation with sums of two squares II

TL;DR

This work advances Diophantine approximation by focusing on integers represented as sums of two squares. It employs a Voronoi summation framework tailored to the quadratic form $Q(x,y)=x^2+y^2$, together with Gauss sums and Poisson summation, to isolate a main term and tightly bound the error term for a smoothed counting function $S$. By carefully balancing parameters and bounding $T_2$ against $T_1$, the authors achieve the exponent $1/2-\varepsilon$ in the sums-of-two-squares setting with a quantitative lower bound, matching the conjectured optimal rate in this special case. The results sharpen prior weaker bounds (e.g., $3/7-\varepsilon$) and provide a robust analytic approach that harmonizes Diophantine approximation with representations by binary quadratic forms, using a blend of spectral-type sums and harmonic analysis techniques.

Abstract

Recently, we showed that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed but arbitrarily small $\varepsilon>0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by weaker exponent $γ<3/7-\varepsilon$. In this article we recover this quantitative version with the exponent $1/2-\varepsilon$, but now for the particular case of sums of two squares.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Corollary 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof