Accumulation points of normalized approximations

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points $q\alpha$ with $\alpha \in {\mathbb{R}}^d$ and $q\in \mathbb{Z}$. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of $\alpha$ whose accumulation points are all of ${\mathbb{R}}^d$. In the second part we focus primarily on the case when the coordinates of $\alpha$ together with $1$ form a basis for an algebraic number field $K$. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when $d=2$, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in $K$) of a single ellipse, or of a pair of hyperbolas, depending on whether or not $K$ has a non-trivial embedding into $\mathbb{C}$.