The discrete Pompeiu problem on the plane

The discrete Pompeiu problem is stemmed from an integral-geometric question on the plane. The problem is whether we can reconstruct a function if we know the average values of the function on every congruent copy of a given pattern. After introducing the theory of spectral analysis on discrete Abelian groups, I show some results for the discrete version of the problem. One of the arguments is connected to a coloring problem of the plane. One of them is a geometric construction and some others based on some geometric and combinatoric properties of the plane. I also mention some unsolved questions of the topic. My talk is based on a joint work with M. Laczkovich and Cs. Vincze.