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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.00173 |
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Table of Contents:
- 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 $\mathbb{C}$.