On pattern entropy of weak model sets

TL;DR

This paper investigates the pattern entropy of weak model sets derived from cut-and-project schemes, providing an upper bound related to the boundary volume of the internal space window, confirming a conjecture by Moody.

Contribution

It establishes a non-trivial upper bound on pattern entropy for weak model sets, advancing understanding of their structural complexity.

Findings

Upper bound on pattern entropy in terms of window boundary volume

Confirmation of Moody's conjecture on pattern entropy

Application to sets like squarefree numbers and visible lattice points

Abstract

We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.