Geometric and arithmetic aspects of approximation vectors

TL;DR

This paper studies the asymptotic distribution of approximation vectors related to Diophantine approximation in higher dimensions, using ergodic theory and adelic space techniques to generalize previous results.

Contribution

It introduces a new framework for analyzing the distribution of approximation vectors in higher dimensions and for algebraic vectors, extending prior work to more general settings.

Findings

Describes limiting measures for approximation vectors for Lebesgue almost every vector.

Establishes equidistribution results for approximation objects in higher dimensions.

Generalizes classical Diophantine approximation results to joint distributions and algebraic settings.

Abstract

Let $\theta\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $\theta$: the projection of the lattice $\mathbb{Z}^{d+1}$ to the hyperplane of the first $d$ coordinates along the approximating vector $(p,q)$; the displacement vector $(p - q\theta)$; and the residue classes of the components of the $(d + 1)$-tuple $(p, q)$ modulo all primes. All of these have been studied in connection with Diophantine approximation problems. We consider the asymptotic distribution of all of these quantities, properly rescaled, as $(p, q)$ ranges over the best approximants and $\epsilon$-approximants of $\theta$, and describe limiting measures on the relevant spaces, which hold for Lebesgue a.e. $\theta$. We also consider a similar problem for vectors $\theta$ whose components, together with 1, span a totally real number field of degree $d+1$. Our technique involve recasting the problem as an equidistribution problem for a cross-section of a one-parameter flow on an adelic space, which is a fibration over the space of $(d + 1)$-dimensional lattices. Our results generalize results of many previous authors, to higher dimensions and to joint equidistribution.