Dimension of the intersection of certain Cantor sets in the plane
In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such tha...
| Published in: | Opuscula Mathematica |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2021-03-01
|
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf |
| _version_ | 1852762361961644032 |
|---|---|
| author | Steen Pedersen Vincent T. Shaw |
| author_facet | Steen Pedersen Vincent T. Shaw |
| author_sort | Steen Pedersen |
| collection | DOAJ |
| container_title | Opuscula Mathematica |
| description | In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane. |
| format | Article |
| id | doaj-art-8e4b39f598cf4e338eefe6b83b4f8b66 |
| institution | Directory of Open Access Journals |
| issn | 1232-9274 |
| language | English |
| publishDate | 2021-03-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| spelling | doaj-art-8e4b39f598cf4e338eefe6b83b4f8b662025-08-19T20:55:28ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742021-03-01412227244https://doi.org/10.7494/OpMath.2021.41.2.2274111Dimension of the intersection of certain Cantor sets in the planeSteen Pedersen0Vincent T. Shaw1Wright State University, Department of Mathematics, 3640 Col Glenn Hwy, Dayton, OH 45435, USAWright State University, Department of Mathematics, 3640 Col Glenn Hwy, Dayton, OH 45435, USAIn this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdfcantor setfractalself-similartranslationintersectiondimensionminkowski dimension |
| spellingShingle | Steen Pedersen Vincent T. Shaw Dimension of the intersection of certain Cantor sets in the plane cantor set fractal self-similar translation intersection dimension minkowski dimension |
| title | Dimension of the intersection of certain Cantor sets in the plane |
| title_full | Dimension of the intersection of certain Cantor sets in the plane |
| title_fullStr | Dimension of the intersection of certain Cantor sets in the plane |
| title_full_unstemmed | Dimension of the intersection of certain Cantor sets in the plane |
| title_short | Dimension of the intersection of certain Cantor sets in the plane |
| title_sort | dimension of the intersection of certain cantor sets in the plane |
| topic | cantor set fractal self-similar translation intersection dimension minkowski dimension |
| url | https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf |
| work_keys_str_mv | AT steenpedersen dimensionoftheintersectionofcertaincantorsetsintheplane AT vincenttshaw dimensionoftheintersectionofcertaincantorsetsintheplane |