The discrete Pompeiu problem on the plane

TL;DR

This paper investigates the discrete Pompeiu property for finite sets in the plane, proving it for parallelograms, rational quadrangles, and certain parametrized quadrangles, with extensions to weighted versions.

Contribution

It establishes the discrete Pompeiu property for various classes of polygons, including parallelograms and rational quadrangles, and explores weighted versions for finite sets.

Findings

Parallelograms have the discrete Pompeiu property w.r.t. isometries.

Quadrangles with rational coordinates have the property.

Finite linear sets with commensurable distances have the weighted property.

Abstract

We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property w.r.t. isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property w.r.t. isometries, and every finite set has the weighted discrete Pompeiu property w.r.t. similarities.