Diophantine approximation with sums of two squares II

Recently, the authors 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 $\varepsilon>0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $γ<3/7-\varepsilon$. In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$, where we confine ourselves to the particular case of sums of two squares.

💡 Research Summary

The paper studies Diophantine approximation on the sparse set of integers that can be written as a sum of two squares, i.e. A = {n = x² + y² : x,y∈ℤ}. For a fixed irrational α, Dirichlet’s theorem guarantees infinitely many n with ‖αn‖ < n⁻¹, but when n is restricted to a thin subset the exponent typically deteriorates. Known results give exponents 1/3 − ε for primes and 2/3 − ε for square‑free numbers. Earlier work of the authors proved the existence of infinitely many n∈A satisfying ‖αn‖ < n^{-(½−ε)} but without any quantitative lower bound.

The present article provides a quantitative version for the exponent ½ − ε in the special case Q(x,y)=x²+y². The authors introduce a weighted sum

S = ∑{b (mod q), (b,q)=1} Φ(b/L) ∑{n≡b (mod q)} r₂(n) w(n/X),

where r₂(n)=4∑_{d|n}χ₄(d) counts representations of n as a sum of two squares, χ₄ is the non‑trivial Dirichlet character modulo 4, Φ and w are non‑negative Schwartz test functions compactly supported in