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arXiv · 2411.14861

Topological structure of the sum of two affine Cantor sets

Mehdi Pourbarat
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Abstract

We introduce a dense subset of affine Cantor sets, termed "generalized homogeneous Cantor sets." We show that for any two members of this class, if their sum set is not a Cantor set, it has a dense interior. If one of them is an affine Cantor set with two mappings, the same result holds. In sequel, we introduce a subclass of generalized homogeneous Cantor sets, denoted by $\cal{N}$, such that for any pair $K, K' \in \cal{N}$, there are generically (i.e., from both topological and measure-theoretic) five possible structures for their sum set: a bilateral gap interval, an L, R, M-Cantorval, or a Cantor set. Finally, we explicitly present new pairs of affine Cantor sets which have stable intersection, while do not satisfy the Generalized Thickness Test.

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Mehdi Pourbarat. 2026-09-19. Topological structure of the sum of two affine Cantor sets. https://arxiv.org/abs/2411.14861

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