Khintchine's Theorem with rationals coming from neighborhoods in different places

TL;DR

This paper extends Khintchine's Theorem to rational approximations sourced from neighborhoods in different completions of the rationals, incorporating Hausdorff measures to generalize Jarník's Theorem.

Contribution

It proves new versions of Khintchine's Theorem involving rationals from different completions and extends results to Hausdorff measures using a mass transference principle.

Findings

Established Khintchine-type theorems with rationals from Euclidean and p-adic completions.

Extended results to Hausdorff measures, generalizing Jarník's Theorem.

Connected approximation properties across different completions of the rationals.

Abstract

The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine's Theorem studied. We prove versions of Khintchine's Theorem where the rational numbers are sourced from a ball in some completion of $\mathbb{Q}$ (i.e. Euclidean or $p$-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarn\'ik's Theorem.