The support of a function with thin spectrum
We prove that if does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty there exists a constant c > 0 such that for all whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.