On the sum of two affine Cantor sets

2024/11/22 by Mehdi Pourbarat, Pourbarat, Mehdi
Computer Science · #28A78 #58F14 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2411.14861

openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that K and K' are two affine Cantor sets. It is shown that the sum set K+K' has equal box and Hausdorff dimensions and in this number named s, H^s(K+K')<∞. Moreover, for almost every pair (K,K') satisfying HD(K)+HD(K')≤ 1, there is a dense subset D⊂ \mathbb R such that H^s(K+λK')=0, for all λ∈ D. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals.

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