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

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JournalErgodic Theory and Dynamical Systems
DateSubmitted - 10 Aug 2016
DateAccepted/In press - 9 Mar 2017
DateE-pub ahead of print (current) - 20 Jun 2017
Number of pages20
Early online date20/06/17
Original languageEnglish

Abstract

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.

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