Small sumsets in real line : a continuous 3k-4 theorem

2016/05/15 by Anne de Roton, de Roton, Anne
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO

paper · pdf · doi:10.48550/arxiv.1605.04597

arxiv created 2016/05/15 · openalex publication_date 2016/05/15 · arxiv updated 2016/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a continuous Freiman's 3k-4 theorem for small sumsets in ℝ by using some ideas from Ruzsa's work on measure of sumsets in ℝ as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when λ(A+B)<λ(A)+λ(B)+min(λ(A),λ(B)). We also give some structural information for sets of large density with small sumset and characterize the extremal sets for which equality holds in the lower bounds for λ(A+B).

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