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

Published online by Cambridge University Press:  20 June 2017

TUSHAR DAS ,
LIOR FISHMAN ,
DAVID SIMMONS  and
MARIUSZ URBAŃSKI
TUSHAR DAS
Affiliation:
University of Wisconsin – La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI 54601, USA email tdas@uwlax.edu
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email lior.fishman@unt.edu, urbanski@unt.edu
DAVID SIMMONS
Affiliation:
University of York, Department of Mathematics, Heslington, York YO10 5DD, UK email David.Simmons@york.ac.uk
MARIUSZ URBAŃSKI
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email lior.fishman@unt.edu, urbanski@unt.edu
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Abstract

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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. 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 Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański 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.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 638 - 657
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge University Press, 2017

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