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

TL;DR

This paper determines the optimal Diophantine exponent for a specific group action on a homogeneous space, showing it is 1 under certain hypotheses, and improves previous bounds significantly.

Contribution

It establishes the exact value of the Diophantine exponent for $ ext{SL}_n(Z[1/p])$ actions, refining prior bounds and removing the need for temperedness assumptions.

Findings

Diophantine exponent lies in [1, 1+O(1/n)]

Exponent equals 1 under Sarnak's density hypothesis

Improves bounds from [1, n-1] to near 1

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.