Published online by Cambridge University Press: 27 November 2019
We study dynamical systems $(X,G,m)$ with a compact metric space
$X$, a locally compact,
$\unicode[STIX]{x1D70E}$-compact, abelian group
$G$ and an invariant Borel probability measure
$m$ on
$X$. We show that such a system has a discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays, for general dynamical systems, a similar role to that of the autocorrelation measure in the study of aperiodic order for special dynamical systems based on point sets.