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Optimal Diophantine Exponents for $\mathrm{SL}(n)$

Subhajit Jana, Amitay Kamber

Abstract

The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series.

Optimal Diophantine Exponents for $\mathrm{SL}(n)$

Abstract

The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the -action on the generalized upper half-space , lies in , substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be . We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local -norms of the Eisenstein series.
Paper Structure (31 sections, 43 theorems, 263 equations, 1 figure)

This paper contains 31 sections, 43 theorems, 263 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Theorems
  3. Almost-covering
  4. Outline of the proof
  5. Generalizations and Open Problems
  6. Structure of the article
  7. Notation
  8. Acknowledgements
  9. The Setting of Ghosh--Gorodnik--Nevo
  10. Diophantine Exponents on the sphere
  11. Preliminaries - Local Theory
  12. Basic Set-up
  13. Spherical Transform
  14. Spherical Representations
  15. Bounds on Spherical Functions
  16. ...and 16 more sections

Key Result

Proposition 1.2

for every $x_0\in \mathbb{H}^n$ and almost every $x\in\mathbb{H}^n$ we have $\kappa(x,x_0)=\kappa(x_0)$, and for almost every $x,x_0\in\mathbb{H}^n$ we have $\kappa(x,x_0)=\kappa(x_0)=\kappa$.

Figures (1)

  • Figure 1: Covering of $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$ by balls around $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{Z}[1/3])$. The height of the points is bounded by $3$, and the balls are of radius $3^{-3}$.

Theorems & Definitions (87)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1: Ghosh--Gorodnik--Nevo, ghosh2018best and \ref{['sec:GGN']}
  • Theorem 2
  • Remark 1.3
  • Theorem 3
  • Theorem 4
  • Remark 1.4
  • Definition 2.1: ghosh2018best, Definition 2.1
  • Remark 2.2
  • ...and 77 more