Decaying and nondecaying badly approximable numbers
Abstract
We call a badly approximable number $decaying$ 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 BanachMazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 DOI:
 10.48550/arXiv.1508.03734
 arXiv:
 arXiv:1508.03734
 Bibcode:
 2015arXiv150803734B
 Keywords:

 Mathematics  Number Theory