The support of a function with thin spectrum
Volume 67 / 1994
Colloquium Mathematicum 67 (1994), 147-154
DOI: 10.4064/cm-67-1-147-154
Abstract
We prove that if $E ⊆ Ĝ$ does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty $S ⊆ G$ there exists a constant c > 0 such that $∥ f1_S ∥_2 ≥ c ∥ f ∥ _2$ for all $f ∈ L^2(G)$ 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.
