Entropy and Diffraction of the $$k$$ k -Free Points in $$n$$ n -Dimensional Lattices

Abstract

Abstract. We consider the kth-power-free points in n-dimensional lattices and explicitly calculate their entropies and diffraction spectra. This is of particular interest since these sets have holes of unbounded inradius.

Toeplitz flows and model sets

“…Indeed, Schlottmann asks whether irregularity of the model set implies failure of unique ergodicity [24] and Moody has suggested that irregularity is related to positive entropy. The suggestion of Moody is recorded in [23] (later subsumed in [22]) and is also discussed in the introduction to [12]. When combined with examples of Toeplitz systems studied in the past, our main theorem allows us to answer these speculations by presenting model sets with various previously unknown features (such as irregularity combined with unique ergodicity and zero entropy).…”

Dynamical Properties of k-Free Lattice Points

“…In Ref. [2], it was shown that the above results remain true for the larger class of kth-power-free (or k-free for short) points of arbitrary lattices in n-space. Furthermore, it was shown there that these sets have positive patch counting entropy but zero measure-theoretical entropy with respect to a measure that is dened in terms of the`tied' frequencies of patches in space.…”

“…Since V ∈ A (otherwise some point of V is in p k Λ for some prime p, a contradiction) and since A is a Λ-invariant and closed subset of {0, 1} Λ , it is clear that X(V ) is a subset of A. By [2], Thm. 2, the other inclusion is also true.…”

“…In view of the binary conguration space interpretation, and following [2], the patch counting entropy of V is dened as…”

“…k-th-power-free) integers (in particular on the ergodicity of the above frequency measure and the dynamical spectrum, but also on the topological dynamics) go beyond what was covered in [2]. For further generalisations in one dimension, see [7,8].…”