(n,k)-Besicovitch sets do not exist in ℤ_pⁿ and ℤⁿ for k≥ 2

2023/12/05 by Manik Dhar, Dhar, Manik
Mathematics · #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2312.02495

openalex publication_date 2023/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Besicovitch showed that a compact set in ℝⁿ which contains a unit line segment in every direction can have measure 0. These constructions also work over other metric spaces like the p-adics and profinite integers. It is conjectured that it is impossible to construct sets with measure 0 which contain a unit 2-disk in every direction in ℝⁿ. We prove that over the p-adics and profinite integers any set which contains a 2-flat in every direction must have positive measure. The main ingredients are maximal Kakeya estimates for (ℤ/Nℤ)ⁿ proven in [Dha22] and adapting Fourier analytic arguments of Oberlin [Obe05]. In general, we prove L^n-1 to L^n-1 estimates for the maximal operator corresponding to 2-flats.

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