Badly approximable points on self-affine sponges and the lower Assouad dimension

TY - JOUR

T1 - Badly approximable points on self-affine sponges and the lower Assouad dimension

AU - Simmons, David Samuel

AU - Das, Tushar

AU - Fishman, Lior

AU - Urbański, Mariusz

N1 - 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 - 2017/6/20

Y1 - 2017/6/20

N2 - We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpi\'nski sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Bara\'nski carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

AB - We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpi\'nski sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Bara\'nski carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

M3 - Article

SN - 0143-3857

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -