Geodesic
Decaying and non-decaying badly approximable numbers
Ryan Broderick1 ; Lior Fishman2 ; David Simmons3
1 University of California, Irvine 340 Rowland Hall (Bldg.# 400) Irvine, CA 92697-3875, U.S.A.
2 Department of Mathematics University of North Texas 1155 Union Circle #311430 Denton, TX 76203-5017, U.S.A.
3 Department of Mathematics University of York Heslington, York YO10 5DD, UK
Acta Arithmetica, Tome 177 (2017) no. 2, pp. 143-152.

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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).
DOI : 10.4064/aa8281-10-2016
Keywords: call badly approximable number decaying roughly lagrange constants integer multiples number decay fast possible terminology question bugeaud asks hausdorff dimension set decaying badly approximable numbers set badly approximable numbers which decaying answer questions showing hausdorff dimensions sets equal part proof utilizes game which combines banach mazur game schmidt game first introduced fishman reams simmons

Ryan Broderick 1 ; Lior Fishman 2 ; David Simmons 3

1 University of California, Irvine 340 Rowland Hall (Bldg.# 400) Irvine, CA 92697-3875, U.S.A.
2 Department of Mathematics University of North Texas 1155 Union Circle #311430 Denton, TX 76203-5017, U.S.A.
3 Department of Mathematics University of York Heslington, York YO10 5DD, UK 
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Ryan Broderick; Lior Fishman; David Simmons. Decaying and non-decaying badly approximable numbers. Acta Arithmetica, Tome 177 (2017) no. 2, pp. 143-152. doi : 10.4064/aa8281-10-2016. http://geodesic.mathdoc.fr/articles/10.4064/aa8281-10-2016/

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