Lacunary self-similar fractal sets and intersection of Cantor sets
type: Item
instance of: scholarly article
title: Lacunary self-similar fractal sets and intersection of Cantor sets
zbMATH Open document ID: 1049.28006
published in: Lobachevskii Journal of Mathematics
publication date: 2003-06-23T00:00:00Z
full work available at URL: https://eudml.org/doc/231456
full work available at URL: igu.htm
review text: Let \(K\) be the Cantor set in \([0,1]\). In this paper the author studies mainly the Hausdorff dimension of the intersection of \(K\) and its translation \(K+a\), where \(a\in [0,1]\). To this end, the author introduces the notion of a lacunary self-similar set. Every self-similar set is lacunary self-similar. It is shown that for almost all \(a\in [0,1]\), the set \(K\cap (K+a)\) is lacunary self-similar and has Hausdorff dimension \(\ln 2 / 3\ln 3\). Finally, this result is generalized to sets of the form \(K^p\cap (K^p+a)\), where \(a\in [0,1]\) and \(K^p\) is the set of numbers in \([0,1]\) that admit base \(p\)-expansions without odd digits.
reviewed by: Jan van Mill
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1932990
zbMATH Keywords: Cantor set
zbMATH Keywords: intersection
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: lacunary self-similar set
author: Konstantin B. Igudesman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1932990