Decaying and non-decaying badly approximable numbers

摘要

We call a badly approximable number $decaying$ if, roughly, the Lagrangeconstants of integer multiples of that number decay as fast as possible. Inthis terminology, a question of Y. Bugeaud ('15) asks to find the Hausdorffdimension of the set of decaying badly approximable numbers, and also of theset of badly approximable numbers which are not decaying. We answer bothquestions, 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 andSchmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).