The Prime times of twisted Diophantine approximation

TL;DR

This paper extends Kurzweil's inhomogeneous Diophantine approximation results to subsets of natural numbers with multiplicative structure, such as primes and sums of squares, and characterizes badly approximable numbers in this context.

Contribution

It generalizes Kurzweil's theorem to structured sets of integers and provides a new characterization of badly approximable numbers based on these sets.

Findings

Extension of Kurzweil's result to primes and sums of two squares

Construction of sets where badly approximable condition is necessary

Improved bounds for approximation for fixed badly approximable numbers

Abstract

The seminal work of Kurzweil (1955) provides for any fixed badly approximable $\alpha$ and monotonically decreasing $\psi$ a Khintchine-type statement on the set of the inhomogeneous real parameters $\gamma$ for which $\lVert n \alpha + \gamma\rVert \leq \psi(n)$ has infinitely many integer solutions, and further shows that the assumption of $\alpha$ being badly approximable is necessary. In this article, we generalize Kurzweil's statement to restricting $n \in \mathcal{A}$, where $\mathcal{A} \subseteq \mathbb{N}$ is a set with some multiplicative structure. We show that for badly approximable $\alpha$, the result of Kurzweil extends to a general class of sets $\mathcal{A}$, which allows us to establish the Kurzweil-type result in particular along the primes and along the sums of two squares. Furthermore, we construct non-trivial sets $\mathcal{A}$ where the assumption of $\alpha$ being badly approximable is necessary. In particular, this criterion applies to $\mathcal{A}$ being the set of square-free numbers, providing a novel characterization of the badly approximable numbers. These statements in particular allow for improving the best known bounds for $\lVert n \alpha + \gamma\rVert \leq \psi(n)$ for infinitely many $n \in \mathcal{A}$ for fixed badly approximable $\alpha$ and for various sets $\mathcal{A}$ of number-theoretic interest when accepting an exceptional set for $\gamma$ of Lebesgue measure $0$.