Decaying and non-decaying badly approximable numbers

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JournalActa Arithmetica
DateIn preparation - 15 Aug 2015
DateAccepted/In press - 10 Oct 2016
DateE-pub ahead of print (current) - 28 Dec 2016
Volume177
Number of pages10
Pages (from-to)143-152
Early online date28/12/16
Original languageEnglish

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 Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).

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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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  • math.NT

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