TITLE: A question about non-norm-euclidean real quadratic fields
QUESTION [22 upvotes]: After lurking MO for a while, I decided I'd jump in.  Why not start with a question?
Specifically, I have a question about a natural set that sort of "measures" the failure of a real quadratic field to be norm-Euclidean.
Let $K$ be a real quadratic field and let $V$ denote the two-dimensional real vector space $K\otimes_{\mathbb Q}\mathbb{R}$.  The ring of integers $\mathcal{O}_K$ embeds as a lattice in $V$, and the quotient is a real torus $T$ equipped with an automorphism of infinite order induced by a fundamental unit.  The field norm on $K$ extends in a natural way to a map $V\to \mathbb{R}$ (extend the galois automorphism and use the usual definition of norm, or just work in coordinates in the obvious fashion).
Now let $U$ be the open set in $V$ consisting of all $y$ such that $|Nm(y-x)|<1$ for some $x\in \mathcal{O}_K$.  Thus, for example, $K$ is norm-Euclidean if $U=V$.  In general the complement is some closed set in $V$ that is $\mathcal{O}_K$-translation invariant and descends to a closed set $X$ of the torus $T$.  
Here's where I wish I knew how to make one of those nifty shaded boxes to highlight the question...

What is the nature of the set $X$?

Though I know basically nothing of ergodic theory, it seems that $X$ must have measure zero since it's preserved by the (ergodic) automorphism induced by the fundamental unit.  It seems to look like some sort of Cantor-like set, but I have no idea how to really get my hands on it.
Two specific questions:
1) Is there some reasonable way to parameterize or otherwise describe $X$ somewhat explicitly (I ran across something called a "binary arithmetic coding" once that purports to be able to do this, but it seemed ill-suited to this particular set to me).
2) What is the Hausdorff dimension of $X$?  Does it have any relation to the arithmetic of $K$?  (Wouldn't it be cute if it had something to do with the class number, etc.)
Of course, one could ask a similar question about any number field other than $\mathbb{Q}$ and quadratic imaginary fields, but this seemed like the place to start.

REPLY [9 votes]: As it stands, I guess your question is hopeless. There are some answers if you replace the condition $|N(x-y)| < 1$ by $|N(x-y)| < \kappa$, where $\kappa$ is chosen as the infimum over all constants $\lambda$ for which $|N(x-y)| < \lambda$ satisfies $U=V$. This infimum is called "Euclidean minimum" (or, classically, the inhomogeneous minimum of the norm-form of the corresponding quadratic number field). 
Barnes and Swinnerton-Dyer (The inhomogeneous minima of binary quadratic forms, Acta Math. 87, 88, 92, 1952--1954) have studied the set X for various families of number fields (and $\kappa$ as above). The set is sometimes finite and sometimes countably infinite. Recent results in this direction have been obtained by Jean-Paul Cerri, Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1 [J. Reine Angew. Math. 592 (2006), 49-62] using ergodic theory.