Decaying and non-decaying badly approximable numbers

TY - JOUR

T1 - Decaying and non-decaying badly approximable numbers

AU - Broderick, Ryan

AU - Fishman, Lior

AU - Simmons, David

N1 - © 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.

PY - 2016/12/28

Y1 - 2016/12/28

N2 - 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).

AB - 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).

KW - math.NT

U2 - 10.4064/aa8281-10-2016

DO - 10.4064/aa8281-10-2016

M3 - Article

SN - 0065-1036

VL - 177

SP - 143

EP - 152

JO - Acta Arithmetica

JF - Acta Arithmetica

ER -