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

TL;DR

The paper resolves the discrete Pompeiu problem for finite sets in ${\mathbb R}^k$ by proving a weighted discrete Pompeiu property with respect to the rigid-motion group $G_k$ for all $k\ge 2$, implying that every finite $K$ with $|K|\ge 2$ is a Jackson set. The authors develop harmonic analysis on countable discrete Abelian groups to obtain key vanishing results (via exponentials and varieties) and apply this to coloring and Steinhaus-tiling problems; they then prove the main result separately for $k=2$ and for $k>2$ with an induction on dimension. Their approach blends discrete harmonic analysis with number-theoretic tools (notably the ESS lemma) and structural combinatorics to derive strong consequences for finite Steinhaus tilings and Jackson-type sets. The work thereby unifies and extends prior results on the discrete Pompeiu problem and yields new rigidity phenomena for colorings and tilings in Euclidean spaces.

Abstract

Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ $(k\ge 2)$. We prove that if a function $f\colon \mathbb{R}^k\to \mathbb{C}$ 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 $\mathbb{R}^k$ having at least two elements is a Jackson set; that is, no subset of $\mathbb{R}^k$ intersects every congruent copy of $K$ in exactly one point.

Theorems & Definitions (15)

• Proposition 1
• Theorem 2
• Theorem 3
• Remark 4
• Corollary 5
• Corollary 6
• Corollary 7
• Corollary 8
• Remark 9
• Remark 10