Accumulation points of normalized approximations

TL;DR

This paper investigates the accumulation points of normalized integer vector translates of points in Euclidean space, revealing measure, dimension, and algebraic structures, especially for points related to algebraic number fields.

Contribution

It provides measure and Hausdorff dimension results for the set of points with all accumulation points, and uncovers algebraic and geometric structures in the case of algebraic number fields.

Findings

Set of points with all accumulation points equals .

Hausdorff dimension results for specific sets.

Structured geometric description for algebraic number field cases.

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 $q\alpha$ 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 $\mathbb{C}$.