Decaying and non-decaying badly approximable numbers

Ryan Broderick, Lior Fishman, David Simmons

Research output: Contribution to journalArticlepeer-review


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 Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).
Original languageEnglish
Pages (from-to)143-152
Number of pages10
JournalActa Arithmetica
Early online date28 Dec 2016
Publication statusE-pub ahead of print - 28 Dec 2016

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© Instytut Matematyczny PAN, 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.


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