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Geometric and arithmetic aspects of approximation vectors

Uri Shapira, Barak Weiss

Abstract

Let $θ\in\mathbb{R}^d$. We associate three objects to each approximation $(p,q)\in \mathbb{Z}^d\times \mathbb{N}$ of $θ$: 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 $(p - qθ)$; 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 $ε$-approximants of $θ$, and describe limiting measures on the relevant spaces, which hold for Lebesgue a.e. $θ$. We also consider a similar problem for vectors $θ$ 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.

Geometric and arithmetic aspects of approximation vectors

Abstract

Let . We associate three objects to each approximation of : the projection of the lattice to the hyperplane of the first coordinates along the approximating vector ; the displacement vector ; and the residue classes of the components of the -tuple 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 ranges over the best approximants and -approximants of , and describe limiting measures on the relevant spaces, which hold for Lebesgue a.e. . We also consider a similar problem for vectors whose components, together with 1, span a totally real number field of degree . 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 -dimensional lattices. Our results generalize results of many previous authors, to higher dimensions and to joint equidistribution.
Paper Structure (45 sections, 70 theorems, 308 equations, 5 figures)

This paper contains 45 sections, 70 theorems, 308 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline of method and structure of the paper
  3. Acknowledgements
  4. A strengthening: the bundle $\mathscr{E}_n$ of lift functionals over projected lattices
  5. Consequences, and prior work
  6. Growth of denominators, the Khinchin-Lévy constant, and a 'Khinchin-Lévy distribution'
  7. Equidistribution of (shapes of) projections in ${\mathscr{X}}_d$
  8. Equidistribution in the bundle of lift functionals
  9. Equidistribution of displacement vectors in $\mathbb{R}^d$
  10. Equidistribution in $\widehat{\mathbb{Z}}^n$, and congruence constraints
  11. Preliminaries
  12. Measurable cross-sections
  13. Tight convergence of measures on lcsc spaces
  14. Equidistribution of visits to a cross-section
  15. Continuity of the cross-section measure construction
  16. ...and 30 more sections

Key Result

Theorem 1.1

For any norm $||\cdot||$ on $\mathbb{R}^d$ there is a probability measure $\mu = \mu_{\operatorname{best},||\cdot||}$ on ${\mathscr{X}}_d \times \mathbb{R}^d \times \widehat{\mathbb{Z}}^n$ such that for Lebesgue almost any $\theta\in\mathbb{R}^d$, the following holds. Let $\mathbf{v}_k\in\mathbb{Z}^ equidistributes with respect to $\mu$. The measure $\mu$ has the following properties: In particul

Figures (5)

  • Figure 1: The set $D_r^{(-\varepsilon,0)}$ bounded by the surface $F_{\varepsilon, r}$ and the disks $D_r$ and $a_{\varepsilon}(D_r)$. Lattice points in $D_r^{(0, \varepsilon)}$ correspond to visits to $\mathcal{S}_{r,<\varepsilon}$.
  • Figure 2: If the cylinder $C_{r(\Lambda)}$ defined by the unique vector $v(\Lambda) \in \Lambda \cap D_{r_0}$ contains another lattice point $w$, then $\Lambda \notin \mathcal{B}$.
  • Figure 3: Defining the norm by a carefully chosen convex set in $\mathbb{R}^2$.
  • Figure 4: The set $\{(x,y): y\in [0,1),\; f_1(y) \leq x \leq f_2(y) \}$ parameterizing $\mathcal{B}$.
  • Figure :

Theorems & Definitions (153)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Corollary 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • ...and 143 more