Inter model sets in $\mathbb R^d$ are model sets

TL;DR

The paper proves that translates of (almost) model sets in Euclidean space remain model sets within modified cut-and-project schemes, preserving key properties of the original window.

Contribution

It establishes that all translates of (almost) model sets are still model sets in adjusted schemes, extending previous understanding.

Findings

Translates of model sets are model sets in modified schemes.

The window's topological and measure-theoretic properties are preserved.

Results apply to a broader class called almost model sets.

Abstract

We show that any translate of a model set is a model set in some modified cut-and-project scheme. Restricting to Euclidean direct space, we show that any translate of an inter model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original window. Our results hold in fact for a class beyond inter model sets, which we call almost model sets.