The discrete Pompeiu problem on the plane

We say that a finite subset $E$ of the Euclidean plane $\R^2$ has
the discrete Pompeiu property with respect to isometries (similarities),
if, whenever $f:\R^2\to \C$ 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 with respect to
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 with respect to
isometries, and every finite set
has the weighted discrete Pompeiu property with respect to similarities.