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.
• Accepted: 2021-01-07.
• Published online: 2021-03-17.