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 language | English |
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Pages (from-to) | 2108-2133 |
Number of pages | 26 |
Journal | International Mathematics Research Notices |
Volume | 2021 |
Issue number | 3 |
DOIs | |
State | Published - 2021 |