Beurling density on expansible locally compact groups and systems of translations

Abstract

In this paper, we investigate some properties of the expansive automorphisms on expansible locally compact groups. We define the upper and lower Beurling densities on expansible locally compact groups. Using this definition, we show that if for some 1<p′<∞, the finite union of translates ∪k=1nTp(fk,Γk) is a p′-Bessel sequence, then the upper Beurling density is finite, and if ∪k=1nTp(fk,Γk) is a (Cq)-system, then the upper Beurling density cannot be finite. In particular, we conclude that there exists no p′-Bessel (Cq)-system in Lp(G) of the form ∪k=1nTp(fk,Γk), where G is an expansible group.