TY  - JOUR
ID  - Pedersen2021
LB  - Pedersen2021
AU  - Pedersen, Steen
AU  - Shaw, Vincent T.
TI  - Dimension of the intersection of certain Cantor sets in the plane
ST  - Dimension of the intersection of certain Cantor sets in the plane
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 2
SP  - 227
EP  - 244
DO  - https://doi.org/10.7494/OpMath.2021.41.2.227
UR  - https://doi.org/10.7494/OpMath.2021.41.2.227
AB  - 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.
KW  - Cantor set
KW  - fractal
KW  - self-similar
KW  - translation
KW  - intersection
KW  - dimension
KW  - Minkowski dimension
SN  - 1232-9274
ER  -