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On the spectral aspect density hypothesis and application

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Pith's one-line read For n >= 4, the paper establishes Sarnak's density hypothesis in the spectral aspect for GL_n(Z) cuspidal representations and uses it to prove the Diophantine exponent of the SL_n(Z[1/p])-action is optimal (kappa = 1).

arxiv 2504.12150 v1 pith:C3PYVIGI submitted 2025-04-16 math.NT

Edgar Assing , Subhajit Jana This is my paper · ORCID
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The reading

Number theory has a famous guess called the Generalized Ramanujan Conjecture, which says that certain building blocks of functions on arithmetic spaces are 'tempered', meaning they have no exceptional growth. A practical replacement is a density hypothesis: even if such exceptions exist, they are very rare. This paper proves that density hypothesis for the cuspidal representations of GL_n(Z) ordered by size of their archimedean parameters, for every n at least 4. It shows the count of representations that are non-tempered at a fixed prime p by amount sigma is bounded by a power of X with a saving that grows with sigma.

The proof uses the Kuznetsov trace formula, a tool that compares spectral data with sums over geometry. The difficult part is constructing a test function whose spectral side is large exactly for the representations of interest. The authors choose the function geometrically, as a smoothed version of a small ball around the identity, and then use microlocal analysis, developed by Nelson and Venkatesh, to control its action on Whittaker vectors. This yields a new lower bound for the relevant Bessel distribution. The geometric side is handled with estimates for orbital integrals, many of which vanish because of support restrictions.

As a consequence, the paper proves a conjecture of Ghosh, Gorodnik, and Nevo: the Diophantine exponent of the action of SL_n(Z[1/p]) on the symmetric space SL_n(R)/SO_n(R) is 1, which is the best possible value.

Extended reading notes

Core claim

Theorem 1: For fixed n >= 4 and prime p, the number of cuspidal representations pi in Omega_cusp(X) with p-adic non-temperedness sigma_p(pi) >= sigma is bounded by O_{p,epsilon}( X^{ (1/2)(n+2)(n-1)(1 - 4 sigma/(n-1)) + epsilon } ). If true, this confirms Sarnak's density hypothesis in the spectral aspect and, via [16, Theorem 3], yields the optimal Diophantine exponent kappa = 1 for the SL_n(Z[1/p])-action on SL_n(R)/SO_n(R).

Load-bearing premise

The proof of the crucial lower bound Proposition 3.1 for the Bessel distribution J_{F_X^sharp, psi}(pi) depends on Lemma 3.2, which asserts the existence of a (g, delta)-localized Whittaker vector W with W(1) >> X^{ n(n-1)/4 (1-delta) } and ||W|| << 1, extracted from Nelson's preprint [23] (Lemmas 7.1 and 7.8, Corollary 17.11, and Theorem 14.12). If those microlocal localization results fail, or are not valid at the stated uniformity, Proposition 3.1 and hence the density bound do not follow.

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Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on established but deep external machinery, especially Nelson's microlocal analysis and the Kuznetsov formula. No new physical or mathematical entities are postulated, and no data-dependent parameters are introduced. The main non-standard input is the localized Whittaker vector construction from the preprint [23].

assumptions (4)
  • domain assumption Microlocal localization results of Nelson [23]: existence of (g, delta)-localized Whittaker vectors (Lemma 3.2, Corollary 17.11) and the operator bounds in Theorem 14.12.
    These deep results from the preprint [23] are the engine behind Proposition 3.1; the present paper does not prove them, only cites and 'extracts' them.
  • standard math Kuznetsov formula on GL_n(Z)\PGL_n(R) as stated in Proposition 2.3.
    Quoted from [14, Prop 2.1]; standard in analytic number theory but a nontrivial input.
  • domain assumption Lower bound for sums of Hecke eigenvalues at p-powers in terms of sigma_p(pi), [14, (4.3)].
    Used in Lemma 5.2 to convert an L^2 bound on lambda_pi into a lower bound for the non-temperedness measure; quoted without proof.
  • standard math Bound L(1, pi, Ad) << ||mu_infinity(pi)||^epsilon from [19].
    Used to remove harmonic weights in Lemma 5.1.

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Pith. "Pith review of On the spectral aspect density hypothesis and application." pith.science (2026). https://pith.science/paper/C3PYVIGI

@misc{pith2026250412150,
  author       = {Pith},
  title        = {Pith review of: On the spectral aspect density hypothesis and application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3PYVIGI}},
  note         = {Machine review of arXiv:2504.12150}
}
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.

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Works this paper leans on

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