Published paper

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

Manfred Einsiedler, G. A. Margulis, Akshay Venkatesh

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Publication metadata

Journal citation: Inventiones Mathematicae 177(1), 137–212 (2009)
Publication date: 2009
DOI: 10.1007/s00222-009-0177-7 ↗
arXiv counterpart: arXiv:0708.4040v1

Abstract

Sections 6.2–6.3 and Appendix C turn spectral gap into the product-group Sobolev matrix-coefficient decay used in Shi Lemma 4.2.

Role in dependence graphs

Proof-critical source

Bounded trajectories of quasi-rays

This paper is included only for the following marked statement:

(6.1), (6.3), (6.9), and (6.10) · §§6.2.3 and 6.3.2–6.3.3; Appendix CConverts spectral gap to a uniform temperedness exponent and then to Sobolev matrix-coefficient decay, including the product-group weak- $\varphi_0$ estimate used in Shi Lemma 4.2.

Proof-critical source

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

This paper is included only for the following marked statements:

Sobolev norms and properties · §§5–6 as cited by BEG20 §2.2Provides the weighted Sobolev framework adopted by BEG20.
(6.1), (6.3), (6.9), and (6.10) · §§6.2.3 and 6.3.2–6.3.3; Appendix CConverts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

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arXiv:0708.4040v1 · explicit fallback for inaccessible version of record

Manfred Einsiedler, G. A. Margulis, Akshay Venkatesh. Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. Inventiones Mathematicae 177 (2009), no. 1, 137–212. Reviewed form: arXiv:0708.4040v1.

Springer's version of record is subscription-controlled. arXiv exposes only the fixed v1 manuscript from August 30, 2007. A January 10, 2009 author-hosted manuscript is materially revised but is not identified as the version of record; it was used only to confirm version changes and mechanically intended corrections. This report therefore evaluates arXiv v1 and does not claim to audit the VOR.

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Generated August 23, 2026

01Statements3 reported findingsCorrect

The effective measure-classification theorem, its topological consequence, and the stated arithmetic application are correct. The proof architecture supplies polynomial control with the advertised dependencies on the ambient groups and lattice; the source-specific notation defects recorded below are mechanical and alter no conclusion.

Theorem 1.3Correct

Effective equidistribution toward an intermediate closed orbit

Pages 3–4 and 22–68 · Theorem 1.3 and Sections 4–16 · arXiv:0708.4040v1

Under the three hypotheses in Section 1.2, the argument alternates an effective ergodic theorem, quantitative additional invariance, and an effective closing dichotomy. Each successful enlargement strictly increases the dimension of the intermediate semisimple subgroup, so the iteration terminates after fewer than $\dim G$ steps. Taking extrema over the finite collection of possible chains makes the final Sobolev degree and exponents depend only on $G$ and $H$ , while the starting threshold may also depend on $\Gamma$ , exactly as stated.

Theorem 1.4Correct

The topological density conclusion follows with a polynomial radius

Pages 5 and 68–69 · Theorem 1.4 and Section 16.3 · arXiv:0708.4040v1

The proof selects a gap among the finitely many possible intermediate-orbit volumes, applies Theorem 1.3 at the upper endpoint, and tests the resulting measure estimate against a bump function of radius $\eta$ around a point of the compact set $F$ . The lower mass bound $\mu_{x_0S}(f_x)\gg\eta^{\dim S}X^{-1}$ dominates the Sobolev error once the auxiliary integer $N$ is sufficiently large. Compactness bounds the height uniformly, and the relation between $X$ and $V$ yields some $\rho>0$ depending only on $G$ and $H$ .

Proposition 17.6Correct

The prehomogeneous-hypersurface counting application

Pages 72–75 · Proposition 17.6 and Sections 17.7.1–17.7.5 · arXiv:0708.4040v1

The bounded-square-part hypothesis forces the projective height of each integral point to grow by a power of $d$ . Proposition 17.4 converts these height bounds into lower bounds for the relevant closed-orbit volumes, including the only possible intermediate orthogonal orbit in Case B. Theorem 1.3 then gives uniform equidistribution, and the standard orbit-to-integral-point translation produces the stated power-saving count. In the $r=8$ calculation, the printed exponent $d^{10}$ must be $d^5$ , as recorded in the proofs section; the ensuing weaker height bound $d^{1/8}$ and the conclusion are unchanged.

02Proofs6 reported findingsCorrect

The material arguments are correct and complete after mechanically correcting the displayed matrix-coefficient moduli, two Appendix B notation slips, and one exponent in the arithmetic application. In particular, the marked spectral-gap-to-Sobolev-decay chain in Sections 6.2–6.3 and Appendix C closes with the stated parameter dependence.

Sections 4–16Correct and complete

The effective classification iteration preserves all dependencies

Pages 22–69 · Sections 4–16 · arXiv:0708.4040v1

The Sobolev calculus in Section 5 controls point evaluation, products, translations, and smoothing. Sections 8–10 turn spectral decay into quantitative genericity and then additional almost-invariance; Sections 11–14 separate the small-periodic-orbit case from effective enlargement; and Section 15 converts almost-invariance on a closed orbit into closeness to its Haar probability. Section 16 iterates only over strictly increasing intermediate dimensions and takes uniform extrema over the resulting finite set of chains, yielding precisely Theorem 1.3.

Equations (6.1), (6.9), and (6.10)Correct and complete

Spectral gap gives the required Sobolev matrix-coefficient decay

Pages 31–35 and 79–81 · Sections 6.2.3, 6.3.2–6.3.3, and Appendix C · arXiv:0708.4040v1

For each almost-simple factor, spectral gap bounds irreducible constituents away from the trivial representation and hence gives a uniform finite $L^p$ exponent, equivalently $1/m$ -temperedness. Cowling–Haagerup–Howe majorization then yields the $\varphi_0^{1/m}$ bound for smooth vectors. For a product, every irreducible constituent has at least one tempered factor; moving the other factors to the second vector preserves the relevant Sobolev norm and gives the weak spherical-function estimate (6.10). Along a one-parameter subgroup with nontrivial projection to every simple factor, $\varphi_0^{\mathrm{wk}}$ decays exponentially, which is exactly the marked input used downstream.

Equations (6.7)–(6.10)Typo

The displayed complex matrix coefficients need modulus signs

Pages 33–34 · Equations (6.7)–(6.10) · arXiv:0708.4040v1

The four bounds print $\langle s.v,w\rangle$ on the left without absolute-value bars, although a complex matrix coefficient is not ordered. The unique correction is $|\langle s.v,w\rangle|$ in each display. The cited majorization theorem, the surrounding phrase “bounds for matrix coefficients,” and every later quantitative use, including the display in Section 12.3, use the modulus. No estimate or downstream exponent changes.

Appendix B polynomial mapTypo

A stray vector symbol appears in the definition of $h(t)$

Page 78 · Appendix B, paragraph beginning “Put $h(t)$ ” · arXiv:0708.4040v1

The manuscript prints $h(t)=\operatorname{Ad}_{u(t)}|_V g'\in\operatorname{SL}(V)$ . The symbol $g'$ has no role and makes the definition ill-formed. Deleting it gives the unique intended polynomial map $h(t)=\operatorname{Ad}_{u(t)}|_V\in\operatorname{SL}(V)$ , which is the map used in every subsequent covolume polynomial. The January 2009 author manuscript makes exactly this correction.

Appendix B intermediate Lie algebraTypo

A tensor-product sign must be a direct-sum sign

Page 79 · Appendix B, contradiction with Lemma 3.4.1 · arXiv:0708.4040v1

After proving that the nilpotent Lie algebra $\mathfrak l$ is $H$ -invariant and intersects $\mathfrak h$ trivially, the manuscript writes that the generated intermediate algebra is $\mathfrak h\otimes\mathfrak l$ . It must be the semidirect vector-space direct sum $\mathfrak h\oplus\mathfrak l$ : invariance gives $[\mathfrak h,\mathfrak l]\subseteq\mathfrak l$ , while trivial intersection makes the sum direct. This is the nonsemisimple intermediate algebra needed for the contradiction, and the January 2009 author manuscript makes exactly this correction.

Section 17.7.5 determinant exponentTypo

The $r=8$ covariant determinant has exponent $5$ , not $10$

Page 75 · Section 17.7.5, first paragraph of the proof in Case B · arXiv:0708.4040v1

The paragraph prints that $\det(\operatorname{cov}(y))$ is a fixed multiple of $d^3$ or $d^{10}$ for $r=7$ or $8$ . Section 17.7.4 immediately establishes the respective identities $\det(\operatorname{cov}(t))\asymp\operatorname{disc}(t)^3$ and $\det(\operatorname{cov}(t))\asymp\operatorname{disc}(t)^5$ , so the unique correction is $d^5$ . The following lower bound $d^{1/8}$ is weaker than the corrected determinant estimate supplies, hence no later claim changes.

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No non-novelty findings.

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Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

Publication metadata

Abstract

Role in dependence graphs

Bounded trajectories of quasi-rays

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

Audit summary

arXiv:0708.4040v1 · explicit fallback for inaccessible version of record

Effective equidistribution toward an intermediate closed orbit

The topological density conclusion follows with a polynomial radius

The prehomogeneous-hypersurface counting application

The effective classification iteration preserves all dependencies

Spectral gap gives the required Sobolev matrix-coefficient decay

The displayed complex matrix coefficients need modulus signs

A stray vector symbol appears in the definition of h(t)h(t)h(t)

A tensor-product sign must be a direct-sum sign

The r=8r=8r=8 covariant determinant has exponent 555, not 101010

A stray vector symbol appears in the definition of $h(t)$

The $r=8$ covariant determinant has exponent $5$ , not $10$