Solutions to the discrete Pompeiu problem and to the finite Steinhaus tiling problem

2024/03/02 by Gergely Kiss, Kiss, Gergely, Miklós Laczkovich +1 · 2 citations
Engineering · Materials Science · Mathematics · #30D05 #3D Shape Modeling and Analysis #43A45 #52C99 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2403.01279

openalex publication_date 2024/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let K be a nonempty finite subset of the Euclidean space ℝ^k (k≥ 2). We prove that if a function f\colon ℝ^k→ ℂ is such that the sum of f on every congruent copy of K is zero, then f vanishes everywhere. In fact, a stronger, weighted version is proved. As a corollary we find that every finite subset K of ℝ^k having at least two elements is a Jackson set; that is, no subset of ℝ^k intersects every congruent copy of K in exactly one point.

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• Bounded common fundamental domains for two lattices
• The Pompeiu problem and spherical spectral analysis

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