Duffin--Schaeffer examples, real residue systems, and Bohr-set primes

TL;DR

This paper generalizes classical approximation theorems by constructing functions with measure-specific approximation properties, utilizing real residue systems and prime distributions in Bohr sets, and extends several foundational results in number theory and harmonic analysis.

Contribution

It introduces a real-valued residue system framework, extends Dirichlet and Rogers theorems to this setting, and analyzes prime distributions and equidistribution in Bohr sets.

Findings

Existence of functions with zero or full measure of approximable numbers based on inhomogeneous parameters.

Extension of Rogers' theorem to real residue systems.

Prime numbers in Bohr sets exhibit similar distribution and equidistribution properties as in classical settings.

Abstract

We prove the following generalization of a well-known result of Duffin and Schaeffer: For any given countable sets $Y \subset\mathbb{R}$ and $Z\subset\mathbb{R}\setminus\operatorname{span}_\mathbb{Q}(\{1\}\cup Y)$, there exist functions $\psi$ such that the set of inhomogeneously $\psi$-approximable numbers has zero measure or full measure, according as the inhomogeneous parameter lies in $Y$ or $Z$. The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Dirichlet's theorem for prime numbers lying in arithmetic progressions to prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov. An appendix by Manuel Hauke answers a combinatorial question that is posed in the introduction.