Gergo Kiss


Title: The discrete Pompeiu problem in $\mathbb{R}^k$ and its consequences

Abstract:  We study the discrete analogue of the integral geometric problem of Dimitrie Pompeiu. We say that a finite subset E of the Euclidean space is Pompeiu, whenever for a given function f the sum of the values of f on any congruent copy of E is zero, then f is identically zero. Although for sets of two or three elements the affirmative answer is easy, until recently, even for four-point sets the answer was not known. Applying harmonic analysis in some varieties connected to the problem and also some results on linear equations of units, we proved that every finite subset of $\mathbb{R}^k$ is Pompieu ($k>2$). The result has a strong consequence to the finite Steinhaus set problem posed by Steve Jackson. We also discuss the connections to problems in Euclidean Ramsey theory.
It is a joint work with Miklós Laczkovich.