Some dynamics in real quadratic fields with applications to inhomogeneous minima

TL;DR

This paper investigates the dynamics of units in real quadratic fields using symbolic coding, revealing properties of inhomogeneous minima and their distribution within the field.

Contribution

It introduces a symbolic coding approach to analyze the action of fundamental units on the torus, connecting dynamics to inhomogeneous minima in quadratic fields.

Findings

Inhomogeneous spectrum of $K$ contains a dense set of elements in $K$

All isolated inhomogeneous minima are elements of $K$

The study links dynamical systems to number field properties

Abstract

Let $K$ be a real quadratic field. We use a symbolic coding of the action of a fundamental unit on the real $2$-torus associated to $K$ to study the family of subsets $X_t$ of norm distance $\geq t$ from the origin. As an application, we prove that inhomogeneous spectrum of $K$ contains a dense set of elements of $K$, and conclude that all isolated inhomogeneous minima lie in $K$.