An Extension of a Theorem of Duffin and Schaeffer in Diophantine Approximation

TL;DR

This paper extends Duffin and Schaeffer's generalization of Khintchine's theorem in metric Diophantine approximation to include rational approximants with numerators and denominators constrained by stronger congruential conditions.

Contribution

It introduces a new extension of the theorem to cases where numerator and denominator are linked by a stronger congruence, broadening the scope of Diophantine approximation results.

Findings

Extended Duffin-Schaeffer theorem to congruential constraints

Maintained validity under stronger numerator-denominator relations

Broadened applicability of metric Diophantine approximation

Abstract

Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended to the case where the numerators and the denominators of the rational approximants are related by a congruential constraint stronger than coprimality.