Some remarks on the Pompeiu problem for groups
David Scott, Alladi Sitaram · Proceedings of the American Mathematical Society · 1988
A Borel set E E in a topological group G G is said to be a P P -set for the space of integrable functions on G G if the zero function is the only integrable function whose integral over all left and right translates of E E by elements of G G is zero. For a "sufficiently nice" group G G and a Borel set E E of finite Haar measure a certain condition on the Fourier transform of a function related to E E is shown to be a sufficient condition for E E to be a P P -set. This condition is then applied to several classes of groups including certain compact groups, certain semisimple Lie groups, the Heisenberg groups and the Euclidean motion group of the plane.