Dynamical Properties of k-Free Lattice Points

Abstract

We revisit the visible points of a lattice in Euclidean n-space together with their generalisations, the k-thpower-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.

Ergodic Properties of Visible Lattice Points

“…Simultaneously, due to the renewed interest in the square-free integers in connection with Sarnak's conjecture, the dynamical sytems point of view became more important, as is obvious from [13,14,31,24]. Here, the focus is more on the dynamical spectrum, which is closely related to the diffraction measure as indicated above, and explained in detail in [5,8].…”

“…Note that this differs from the definition in [31,24], where, in view of the binary configuration space interpretation, we used the base 2 logarithm. It can be shown by a classic subadditivity argument that this limit exists.…”

Ergodic properties of visible lattice points

“…Simultaneously, due to the renewed interest in the square-free integers in connection with Sarnak's conjecture, the dynamical systems point of view became more important, as is obvious from [13,14,31,24]. Here, the focus is more on the dynamical spectrum, which is closely related to the diffraction measure as indicated above and explained in detail in [5,8].…”

“…This also implies that (X V , Z 2 ) has trivial topological point spectrum (see [24] for an alternative argument that the nonzero constant function is the only continuous eigenfunction of the translation action).…”

“…In terms of the metric d on X V [38,31,24] this means that d(X + t, Y + t) ≤ 1/ρ and the proximality of the system follows. Similarly, the assertion on the unique Z 2 -minimal subset {∅} follows from the fact that any element of X V contains arbitrarily large holes and thus any nonempty closed subsystem contains ∅.…”