On the spectral aspect density hypothesis and application

2025/04/16 by Edgar Assing, Subhajit Jana, Assing, Edgar +1
Mathematics · #11F72 #11L05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2504.12150

openalex publication_date 2025/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the density of non-tempered (at any p-adic place) cuspidal representations for GL_n(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 SL_n(ℤ[1/p])-action on SL_n(ℝ)/SO_n(ℝ) is optimal -- resolving a conjecture of Ghosh, Gorodnik, and Nevo.

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