On the spectral aspect density hypothesis and application
Edgar Assing, Subhajit Jana
TL;DR
This work proves a density hypothesis for the archimedean spectral family $\Omega_{\mathrm{cusp}}(X)$ of cuspidal automorphic representations on $\mathrm{PGL}_n(\mathbb{Z})\backslash \mathrm{PGL}_n(\mathbb{R})$, showing that highly non-tempered representations at a fixed prime $p$ are rare: for $n\ge 4$ and $\sigma\ge 0$, the count of $\pi$ with $\sigma_p(\pi)\ge \sigma$ in $\Omega_{\mathrm{cusp}}(X)$ satisfies $\#\{\pi: \sigma_p(\pi)\ge \sigma\} \ll_{p,\varepsilon} X^{\tfrac{1}{2}(n+2)(n-1)\left(1-\tfrac{4\sigma}{n-1}\right)+\varepsilon}$. The authors develop an archimedean analogue of the non-archimedean principal-congruence approach, leveraging microlocal analysis in the Nelson–Venkatesh orbit framework to construct a test function $F_X$ localized near the identity and a Bessel distribution $J_{\pi,\boldsymbol{\psi}_X}(F_X)$ with a robust spectral lower bound. They combine this with a Kuznetsov-type trace formula to control the geometric side, ultimately translating the spectral bound into a density statement and, as an application, proving the optimal Diophantine exponent $\kappa=1$ for the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$ for all $n\ge 4$. This confirms Sarnak’s density philosophy in this setting and resolves conjectures about optimal lifting in the Diophantine approximation problem on homogeneous spaces. The methods fuse spectral theory of automorphic forms with microlocal orbit methods, advancing the understanding of temperedness distribution in higher-rank archimedean spectra.
Abstract
We prove that the density of non-tempered (at any $p$-adic place) cuspidal representations for $\mathrm{GL}_n(\mathbb{Z})$, varying over a family of representations ordered by their infinitesimal characters, is small -- confirming Sarnak's density hypothesis in this set-up. Among other ingredients, the proof uses tools from microlocal analysis for Lie group representations as developed by Nelson and Venkatesh. As an application, we prove that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$ is \emph{optimal} -- resolving a conjecture of Ghosh, Gorodnik, and Nevo.