Turan's extremal problem on locally compact abelian groups

Abstract

Let G be a locally compact abelian group (LCA group) and U be an open, 0-symmetric set. Let F:=F(U) be the set of all real valued continuous functions from G to R which are supported in U and are positive definite. The Turan constant T(U) of U is then defined as the supremum of the integral of any f on U, which belongs to F and is normalised to have f(0)=1. Mihalis Kolountzakis and the author has shown that structural properties - like spectrality, tiling or packing with a certain set L - of subsets U in finite or compact groups and on R^d and in Z^d, yield estimates of T(U). However, in these estimates some notion of the size, i.e. density of L, played a natural role, and thus in groups where we had no grasp of the notion, we could not accomplish such estimates. In the present work recent new notions of uniform asymptotic upper density are invoked, allowing a more general investigation of the Turan constant in relation to the above structural properties. Our main result extends a result of Arestov and Berdysheva, (also obtained independently and along different lines by Kolountzakis and the author), stating that convex tiles of a Euclidean space necessarily have T(U) =|U|/2^{d}. In our extension R^d could be replaced by any LCA group, convexity is dropped, and the condition of tiling is also relaxed to a certain packing type condition and positive uniform asymptotic upper density of the set L.