Decaying and non-decaying badly approximable numbers
Number Theory
2016-04-20 v2
Abstract
We call a badly approximable number if, roughly, the Lagrange constants of integer multiples of that number decay as fast as possible. In this terminology, a question of Y. Bugeaud ('15) asks to find the Hausdorff dimension of the set of decaying badly approximable numbers, and also of the set of badly approximable numbers which are not decaying. We answer both questions, showing that the Hausdorff dimensions of both sets are equal to one. Part of our proof utilizes a game which combines the Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).
Keywords
Cite
@article{arxiv.1508.03734,
title = {Decaying and non-decaying badly approximable numbers},
author = {Ryan Broderick and Lior Fishman and David Simmons},
journal= {arXiv preprint arXiv:1508.03734},
year = {2016}
}