2020/06/25 by André P. Oliveira, Oliveira, Andre P.
Mathematics · #advanced mathematical theories #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2006.14764
The Duffin--Schaeffer Conjecture answers a question on how well one can\napproximate irrationals by rational numbers in reduced form (an imposed\ncondition) where the accuracy of the approximation depends on the rational\nnumber. It can be viewed as an analogue to Khintchine's Theorem with the added\nrestriction of only allowing rationals in reduced form. Other conditions such\nas numerator or denominator a prime, a square-free integer, or an element of a\nparticular arithmetic progression, etc. have also been imposed and analogues of\nKhintchine's Theorem studied. We prove versions of Khintchine's Theorem where\nthe rational numbers are sourced from a ball in some completion of \ℚ\n(i.e. Euclidean or p-adic), while the approximations are carried out in a\ndistinct second completion. Finally, by using a mass transference principle for\nHausdorff measures, we are able to extend our results to their corresponding\nanalogues with Haar measures replaced by Hausdorff measures, thereby\nestablishing an analogue of Jarn 'ik's Theorem.\n