Rational approximation and arithmetic progressions

TL;DR

This paper develops a comprehensive theory for approximating irrationals by rational fractions with numerators and denominators in specific arithmetic progressions, introducing a Khintchine type theorem for uniform approximation and exploring various perspectives.

Contribution

It introduces a novel Khintchine type theorem for uniform approximation within the context of arithmetic progressions, expanding the understanding of rational approximation methods.

Findings

Established a Khintchine type theorem for uniform approximation

Provided metrical and non-metrical results on approximation quality

Discussed applications of the approximation theory

Abstract

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.