Some symbolic dynamics in real quadratic fields with applications to inhomogeneous minima

Abstract

Let K be a real quadratic field. We use a symbolic coding of the action of a fundamental unit on the torus $T_K=\left(K{\otimes }_{Q}R\right)/O_K$ to study the family of subsets $X_t\subseteq T_K$ of norm distance ≥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.