Accumulation points of normalized approximations

Abstract

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 $qα$ with $α\in\mathbb{R}^d$ and $q\in\mathbb{Z}$. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of $α$ whose accumulation points are all of $\mathbb{R}^d$. In the second part we focus primarily on the case when the coordinates of $α$ 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 $\mathbb{C}$.

Figures (2)

• Figure 1: $(1/2)$-normalized approximations to $\bm{\alpha}=(2^{1/3},2^{2/3})$ (left) and to $\bm{\alpha}=(\alpha,\alpha^2)$, where $\alpha$ is the positive root of $x^3+x^2-2x-1$ (right).
• Figure 2: $(1/2)$-normalized approximations to $\bm{\alpha}=(\alpha,\alpha^2)$, where $\alpha$ is the smallest positive root of $x^3-19x+21$.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8
• Theorem 9
• Lemma 10