Opuscula Math. 41, no. 2 (2021), 227-244

https://doi.org/10.7494/OpMath.2021.41.2.227

Opuscula Mathematica

# Dimension of the intersection of certain Cantor sets in the plane

Steen Pedersen

Vincent T. Shaw

Abstract. 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.

Keywords: Cantor set, fractal, self-similar, translation, intersection, dimension, Minkowski dimension.

Mathematics Subject Classification: 28A80, 51F99.

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- Steen Pedersen (corresponding author)
- Wright State University, Department of Mathematics, 3640 Col Glenn Hwy, Dayton, OH 45435, USA

- Vincent T. Shaw
- Wright State University, Department of Mathematics, 3640 Col Glenn Hwy, Dayton, OH 45435, USA

- Communicated by Palle E.T. Jorgensen.

- Received: 2020-12-26.
- Accepted: 2021-01-07.
- Published online: 2021-03-17.