  With a view to establishing measure theoretic approximation properties of
Delone sets, we study a setup which arises naturally in the problem of
averaging almost periodic functions along exponential sequences. In this
setting, we establish a full converse of the Borel-Cantelli lemma. This
provides an analogue of more classical problems in the metric theory of
Diophantine approximation, but with the distance to the nearest integer
function replaced by distance to an arbitrary Delone set.
