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Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets

2008/07/23 by Pablo Shmerkin, Shmerkin, Pablo
Computer Science · Mathematics · #28A80 #37C45 #Advanced Topology and Set Theory #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.CA #math.DS #msc:28A80 #msc:37C45

paper · pdf · doi:10.48550/arxiv.0807.3709

15 pages. Expository notes, will not be submitted

arxiv created 2008/07/23 · openalex publication_date 2008/07/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller.

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