A "rare'' plane set with Hausdorff dimension 2

2019/04/18 by Vladimir Eiderman, Michael Larsen, Eiderman, Vladimir +1
Mathematics · #28A78 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #math.CA #msc:28A78

paper · pdf · doi:10.48550/arxiv.1904.09034

7 pages

openalex publication_date 2019/04/18 · arxiv created 2019/08/03 · arxiv updated 2019/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for every at most countable family \f_k(x)\ of real functions on [0,1) there is a single-valued real function F(x), x∈[0,1), such that the Hausdorff dimension of the graph Γ of F(x) equals 2, and for every C∈\mathbb R and every k, the intersection of Γ with the graph of the function f_k(x)+C consists of at most one point. We also construct a family of functions of cardinality continuum and a function F with similar properties.

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