Balayage by Fourier transforms with sparse frequencies in compact abelian torsion groups

Abstract:

Let $\Lambda$ be a discrete subset of a LCA group and E a compact subset of the dual group $\Gamma$. Balayage is said to be possible for ($\Lambda$, E) if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by $\Lambda$. For a class of infinite compact metrizable $\Gamma$, including all such torsion groups, we show how to construct $E \subset \Gamma$ such that there are arbitrarily sparse sets $\Lambda$ with balayage possible for ($\Lambda$, E). E is, moreover, large enough that the set of products $E \cdot E \cdot E = \Gamma$.