Large subsets of Euclidean space avoiding infinite arithmetic progressions

2022/05/10 by Laurestine Bradford, Bradford, Laurestine, Hannah Kohut +3
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2205.04786

openalex publication_date 2022/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that if a subset of ℝ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each λ in [0,1), we construct a subset of ℝ that intersects every interval of unit length in a set of measure at least λ, but that does not contain any infinite arithmetic progression.

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