Hausdorff dimension in inhomogeneous diophantine approximation

Yann Bugeaud, Dong Han Kim, Seonhee Lim, Michal Rams

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Let α be an irrational real number. We show that the set of ϵ-badly approximable numbers has full Hausdorff dimension for some positive ϵ if and only if α is singular on average. The condition is equivalent to the average 1/k Σ i=1,••• ,k log ai of the logarithms of the partial quotients ai of α going to infinity with k. We also consider one-sided approximation, obtain a stronger result when ai tends to infinity, and establish a partial result in higher dimensions.

Original languageEnglish
Pages (from-to)2108-2133
Number of pages26
JournalInternational Mathematics Research Notices
Volume2021
Issue number3
DOIs
StatePublished - 2021

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