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

David Samuel Simmons, Tushar Das, Lior Fishman, Mariusz Urbański

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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.
Original languageEnglish
Number of pages20
JournalErgodic Theory and Dynamical Systems
Early online date20 Jun 2017
Publication statusE-pub ahead of print - 20 Jun 2017

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