TY - JOUR

T1 - Schmidt's game, fractals, and numbers normal to no base

AU - Broderick, Ryan

AU - Bugeaud, Yann

AU - Fishman, Lior

AU - Kleinbock, Dmitry

AU - Weiss, Barak

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Given b > 1 and y ∈ R/Z, we consider the set of x ∈ R such that y is not a limit point of the sequence {bnx mod 1 : n ∈ N}. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with 'sufficiently regular' fractals K ⊂ R (that is, supporting measures μ satisfying certain decay conditions). Furthermore, the intersection has full dimension in K if μ satisfies a power law (this holds for example if K is the middle third Cantor set). Thus it follows that the set of numbers in the middle third Cantor set which are normal to no base has dimension log 2/ log 3.

AB - Given b > 1 and y ∈ R/Z, we consider the set of x ∈ R such that y is not a limit point of the sequence {bnx mod 1 : n ∈ N}. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with 'sufficiently regular' fractals K ⊂ R (that is, supporting measures μ satisfying certain decay conditions). Furthermore, the intersection has full dimension in K if μ satisfies a power law (this holds for example if K is the middle third Cantor set). Thus it follows that the set of numbers in the middle third Cantor set which are normal to no base has dimension log 2/ log 3.

UR - http://www.scopus.com/inward/record.url?scp=77952078678&partnerID=8YFLogxK

U2 - 10.4310/mrl.2010.v17.n2.a10

DO - 10.4310/mrl.2010.v17.n2.a10

M3 - Article

AN - SCOPUS:77952078678

SN - 1073-2780

VL - 17

SP - 309

EP - 323

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 2

ER -