Real Analysis Exchange

The Structure of Arithmetic Sums of Affine Cantor Sets

Razvan Anisca, Christopher Chlebovec, and Monica Ilie

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Abstract

In this paper we describe the structure of the arithmetic sum of two affine Cantor sets. These are self-similar sets which are part of the dynamically defined Cantor sets. Let \({\textbf{C}_1}, {\textbf{C}_2}\) be affine Cantor sets with \([0,s]\) and \([0,r]\) as intervals of step 0. We show a generic family of these self-similar sets for which the structure of \({\textbf{C}_1}+{\textbf{C}_2}\) is of one of the following five types: (i) an \(M\)-Cantorval, (ii) an \(R\)-Cantorval, (iii) an \(L\)-Cantorval, or there exist \(\lambda, \eta >0\) and intervals \(I\), \({\tilde I}\) of the construction of \({\textbf{C}_1}\) and \({\textbf{C}_2}\), respectively, such that (iv) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I} \,= \,[0, \lambda s + \eta r]\),\, or (v) \(\lambda {\textbf{C}_1}+\eta {\textbf{C}_2}= {\textbf{C}_1}\cap I + {\textbf{C}_2}\cap {\tilde I} -\min I - \min {\tilde I}\) is homeomorphic to the Cantor ternary set. This result generalizes the description obtained by Mendes and Oliveira for the case of homogeneous Cantor sets and the one obtained by the first two authors for semi-homogeneous Cantor sets. It also provides a suitable framework in which a question of Mendes and Oliveira admits a positive answer.

Article information

Source
Real Anal. Exchange, Volume 37, Number 2 (2011), 325-332.

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1366030628

Mathematical Reviews number (MathSciNet)
MR3080595

Zentralblatt MATH identifier
1263.46010

Subjects
Primary: 28A80: Fractals [See also 37Fxx]
Secondary: 11B05: Density, gaps, topology

Keywords
Cantor set sums of sets

Citation

Anisca, Razvan; Chlebovec, Christopher; Ilie, Monica. The Structure of Arithmetic Sums of Affine Cantor Sets. Real Anal. Exchange 37 (2011), no. 2, 325--332. https://projecteuclid.org/euclid.rae/1366030628


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