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Decaying and non-decaying badly approximable numbers

Volume 177 / 2017

Ryan Broderick, Lior Fishman, David Simmons Acta Arithmetica 177 (2017), 143-152 MSC: Primary 11J04; Secondary 11J06, 11J70, 28A78. DOI: 10.4064/aa8281-10-2016 Published online: 28 December 2016


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 (2015) 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 $1$. Part of our proof utilizes a game which combines the Banach–Mazur game and Schmidt’s game, first introduced in Fishman, Reams, and Simmons (2016).


  • Ryan BroderickUniversity of California, Irvine
    340 Rowland Hall (Bldg.# 400)
    Irvine, CA 92697-3875, U.S.A.
  • Lior FishmanDepartment of Mathematics
    University of North Texas
    1155 Union Circle #311430
    Denton, TX 76203-5017, U.S.A.
  • David SimmonsDepartment of Mathematics
    University of York
    Heslington, York YO10 5DD, UK

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