Some dynamics in real quadratic fields with applications to inhomogeneous minima

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 $\ge t$ 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 $K$.