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Turán’s extremal problem on locally compact abelian groups. (English) Zbl 1240.22005
Summary: Let \(G\) be a locally compact abelian group (LCA group) and \(\Omega\) be an open, 0-symmetric set. Let \(\mathcal F := \mathcal F(\Omega)\) be the set of all continuous functions \(f : G \to \mathbb R\) which are supported in \(\Omega\) and are positive definite. The Turán constant of \(\Omega\) is defined as \[ \mathcal T (\Omega) := \sup \big\{\int_\Omega f : f \in \mathcal F(\Omega), f(0) = 1\big\}. \] M. Kolountzakis and the author have shown that structural properties – like spectrality, tiling or packing with a certain set \(\Lambda\) – of subsets \(\omega\) in finite, compact or Euclidean (i.e., \(\mathbb R^d\)) groups and in \(\mathbb Z^d\) yield estimates of \(\mathcal T (\Omega)\). However, in these estimates some notion of the size, i.e., density of \(\Lambda\) 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, a recent generalized notion of asymptotic uniform upper density is invoked, allowing a more general investigation of the Turán constant in relation to the above structural properties. Our main result extends earlier results stating that convex tiles of a Euclidean space necessarily have \[ \mathcal T_{\mathbb R^d}(\Omega) = | \Omega| /2^d. \] In our extension, \(\mathbb R^d\) could be replaced by any LCA group, convexity is considerably relaxed to \(\Omega\) being a difference set, and the condition of tiling is also relaxed to a certain packing type condition and positive asymptotic uniform upper density of the set \(\Lambda\).
Also, our goal is to give a more complete account of all the related developments and the history, because until now an exhaustive overview of the full background of the so-called Turán problem has not been delivered.

MSC:
22B10 Structure of group algebras of LCA groups
43A35 Positive definite functions on groups, semigroups, etc.
05B40 Combinatorial aspects of packing and covering
11K70 Harmonic analysis and almost periodicity in probabilistic number theory
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