Point Sets and Dynamical Systems in the Autocorrelation Topology

This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced. The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a G-dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a Gdynamical system, is a factor of the dynamical local hull. ∗RVM is grateful for the continuing support of the Natural Sciences and Engineering Research Council of Canada. †Also research assistant at the Institute of Mathematics “Simon Stoilow”, Bucharest, Romania