Values of quadratic forms on quasicrystals and a related problem

TL;DR

This paper investigates the density of quadratic form values on quasicrystals using homogeneous dynamics, establishing conditions for density and classifying orbit closures, with applications to integral values of quadratic and linear forms.

Contribution

It introduces new conditions for density of quadratic form values on quasicrystals and classifies orbit closures in this context, advancing the understanding of their structure.

Findings

Set of quadratic form values can be dense under certain conditions.

Orbit closure classification is achieved for the action on cut and project sets.

Applications include analyzing integral values of quadratic and linear forms.

Abstract

In this paper we study the set of values of quadratic form at points of a cut and project set. We will establish conditions which ensure that the set of values is dense. Our methods involve homogeneous dynamics and we will prove a orbit closure classification type result in this setting. This result has additional applications. In particular, we use it to study the set of integral values of a system consisting of a quadratic form and several linear forms.