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Dimension of the intersection of certain Cantor sets in the plane

Pedersen Steen, Staw Vincent T.

Opuscula Mathematica

2021

Vol. 41, no. 2

227--244

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 Fβ be the subset of F consisting of all x such that the dimension of the intersection of T with its translate by x is β times the dimension of T. We find conditions on the retained digits sets under which Fβ is dense in F for all 0 ≤ β ≤ 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.