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

PDF Version Also Available for Download.

Description

This article highlights 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. The 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.

Physical Description

20 p.

Creation Information

Das, Tushar; Fishman, Lior; Simmons, David & Urbański, Mariusz June 20, 2017.

Context

This article is part of the collection entitled: UNT Scholarly Works and was provided by the UNT College of Science to the UNT Digital Library, a digital repository hosted by the UNT Libraries. More information about this article can be viewed below.

Who

People and organizations associated with either the creation of this article or its content.

Authors

Unknown Creator Role

Author

Provided By

UNT College of Science

The College of Science provides students with the high-demand skills and knowledge to succeed as researchers and professionals. The College includes four departments: Biology, Chemistry, Math, and Physics, and is also home to a number of interdisciplinary programs, centers, institutes, intercollegiate programs, labs, and services.

Contact Us

What

Descriptive information to help identify this article. Follow the links below to find similar items on the Digital Library.

Degree Information

Description

This article highlights 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. The 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.

Physical Description

20 p.

Notes

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. 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.

Source

  • Ergodic Theory and Dynamic Systems, 39(3), Cambridge University Press, June 20, 2017, pp. 1-20

Language

Item Type

Identifier

Unique identifying numbers for this article in the Digital Library or other systems.

Publication Information

  • Publication Title: Ergodic Theory and Dynamic Systems
  • Volume: 39
  • Issue: 3
  • Peer Reviewed: Yes

Collections

This article is part of the following collection of related materials.

UNT Scholarly Works

Materials from the UNT community's research, creative, and scholarly activities and UNT's Open Access Repository. Access to some items in this collection may be restricted.

What responsibilities do I have when using this article?

When

Dates and time periods associated with this article.

Creation Date

  • June 20, 2017

Added to The UNT Digital Library

  • March 3, 2020, 10:24 p.m.

Description Last Updated

  • Nov. 30, 2023, 11:08 a.m.

Usage Statistics

When was this article last used?

Yesterday: 0
Past 30 days: 0
Total Uses: 8

Interact With This Article

Here are some suggestions for what to do next.

Start Reading

PDF Version Also Available for Download.

International Image Interoperability Framework

IIF Logo

We support the IIIF Presentation API

Das, Tushar; Fishman, Lior; Simmons, David & Urbański, Mariusz. Badly approximable points on self-affine sponges and the lower Assouad dimension, article, June 20, 2017; (https://digital.library.unt.edu/ark:/67531/metadc1616621/: accessed June 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Science.

Back to Top of Screen