Turan's extremal problem on locally compact abelian groups

TL;DR

This paper extends the understanding of Turan's extremal problem on locally compact abelian groups by introducing new density notions and relaxing geometric conditions, connecting structural properties with bounds on the Turan constant.

Contribution

It generalizes previous results by replacing Euclidean space with any LCA group, removing convexity, and relaxing tiling conditions using uniform asymptotic upper density.

Findings

Convex tiles in Euclidean space have Turan constant |U|/2^d.

The results extend to general LCA groups with relaxed geometric conditions.

New density concepts enable broader applications in extremal harmonic analysis.

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.