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Journal citation
International Mathematics Research Notices 2021(9), 7060–7095 (2021)
Publication date
2021
arXiv counterpart
arXiv:1510.05256v7
MSC 2020
28A33, 37C85, 22E40, 17B45
As specified in the publication (MSC 2010)
28A33, 37C85, 22E40, 17B45 ↗Full author manuscript, first page (primary 28A33, 37C85; secondary 22E40, 17B45)All four codes carry forward unchanged in the official conversion table.
Official MSC conversion
MSC 2010 → MSC 2020 ↗

Abstract

The paper defines the expanding cone associated with a horospherical subgroup and proves quantitative nonescape of mass and equidistribution results for expanding translates of horospherical slices on finite-volume homogeneous spaces.

Role in dependence graphs

Proof-critical source

Bounded trajectories of quasi-rays

This paper is included only for the following marked statements:

  • Theorem 1.2 and §2.3 · pp. 2–3, 14Algebraic and explicit descriptions of the expanding cone.
  • Theorem 1.5 ($k=1$) · p. 6, (1.7)–(1.10); proof pp. 25–27, Theorem 4.5 specialized to $k=1$Effective equidistribution of a horospherical slice. For the focal specialization, the only analytic branches are quantitative nonescape at (4.16) and spectral-gap thickening at (4.18); the later induction steps for multiple correlations are outside the marked scope.
  • Theorem 1.3 · p. 4, (1.5); proof §§3.1–3.3, pp. 16–22Quantitative nonescape of mass for expanding UU-slices. It bounds the bad slice set in (4.16), one of the two proof-critical inputs to the k=1k=1 case of Theorem 1.5.
  • Lemma 4.1 and Haar decomposition (4.1) · p. 22Provides the stable-central-unstable conull cell and integration formula used to thicken a G+G^+-slice.
  • Lemma 4.2 · p. 23Turns the assumed spectral gap into exponential Sobolev matrix-coefficient decay along btb_t when every simple-factor projection of bb is nontrivial.
  • Lemmas 4.3–4.4 · pp. 23–24, especially (4.2)–(4.3); application at (4.18), p. 27Thickens a good G+G^+-slice with a controlled bump function and combines it with Lemma 4.2 to produce the good-slice estimate (4.18).

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:

  • Theorem 1.5 specialized to k=1 · arXiv:1510.05256v7 p. 6, (1.7)–(1.10); proof pp. 25–27, Theorem 4.5Effective equidistribution of one translated horospherical slice, supplying BG23's EQ1 input.
  • Theorem 1.3 · arXiv v7 p. 4; proof §§3.1–3.3, pp. 16–22Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.
  • Lemma 4.1 and formula (4.1) · arXiv v7 p. 22Stable-central-unstable conull cell and Haar decomposition for thickening.
  • Lemma 4.2 · arXiv v7 p. 23Converts spectral gap to exponential Sobolev matrix-coefficient decay along the chosen diagonal element.
  • Lemmas 4.3–4.4 · arXiv v7 pp. 23–24, especially (4.2)–(4.3); application (4.18), p. 27Controlled bump-function thickening and its Sobolev loss.

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Version of record · International Mathematics Research Notices 2021 (2021), no. 9, 7060–7095 · public full-text HTML

Ronggang Shi. Expanding Cone and Applications to Homogeneous Dynamics. International Mathematics Research Notices 2021 (2021), no. 9, 7060–7095.

Open audited source ↗
Generated August 23, 2026
01Statements4 reported findingsCorrect

The explicit expanding-cone theorem, quantitative nonescape on semisimple quotients, ineffective equidistribution, effective multiple equidistribution, and the ergodic consequences are correct at their stated scope. The proof of Theorem 1.3 contains a finite-index error, but replacing one exponent by a factorial repairs it and consequently preserves Theorem 1.5.

Theorems 1.2 and 1.4Correct

The cone characterization supplies the divergence condition needed for equidistribution

Version of record · Theorem 1.2, Lemmas 2.1–2.9, and Theorem 1.4

The dual-cone argument identifies AU+A_U^+ with exp⁡(au+)\exp(\mathfrak a_{\mathfrak u}^+) by testing every nontrivial irreducible representation on its UU-fixed weights. For a sequence drifting away from every wall, each nonzero UU-fixed weight tends to infinity. That is precisely the C(ρ⊕ρˉ)C(\rho\oplus\bar\rho)-divergence condition (3.13) in the exact Shah–Weiss proof, so the limiting homogeneous measure in Theorem 1.4 follows. The numerical slips in the worked SL⁡m+n\operatorname{SL}_{m+n} example do not enter this general proof.

IMRN version of record ↗
Theorem 1.3Correct after verified repair

Quantitative nonescape is valid on the closed expanding cone

Version of record · Theorem 1.3 and Section 3, especially Lemmas 3.2, 3.3, and 3.7

For arithmetic factors, polynomial goodness and exterior-power quantitative nondivergence give a uniform power of ε\varepsilon. For rank-one factors, Buenger–Zheng’s one-parameter estimate is uniform in the unipotent direction, and polar integration gives the same type of estimate on a UU-ball. The lattice decomposition then reduces the general semisimple quotient to these factors. Lemma 3.7 incorrectly raises a coset representative to the subgroup index; using the factorial of that index, as detailed in the proof finding, restores the injectivity-radius comparison. All estimates remain uniform for a∈AU+‾a\in\overline{A_U^+} because the relevant UU-fixed weights are at least one on the closed cone.

IMRN version of record ↗
Theorem 1.5Correct after verified ancestor repairs

The effective multiple-equidistribution induction closes after the nonescape repair

Version of record · Lemmas 4.1–4.4, Theorem 4.5, and proof of Theorem 1.5

The spectral-gap estimate gives exponential matrix-coefficient decay along btb_t because b1b_1 has a nontrivial projection to every simple factor of HH. Thickening an Hb+H_b^+-slice produces a mixing error r−3ℓe−tδ0r^{-3\ell}e^{-t\delta_0}, while Theorem 1.3 bounds the cusp part by rδ1r^{\delta_1}. The choice r=exp⁡(−tδ0/(3ℓ+δ1))r=\exp(-t\delta_0/(3\ell+\delta_1)) balances the two. Iterating Theorem 4.5 over the cumulative parameters yields (1.10), since every consecutive product has depth at least ⌊a⌋\lfloor\mathbf a\rfloor. The audited defects in Buenger–Zheng and Kleinbock–Margulis have verified repairs and do not invalidate the exact imported statements used here.

IMRN version of record ↗
Theorem 1.6, Corollary 1.7, and Theorem A.1Correct

The periodic and pointwise consequences follow from the effective estimate

Version of record · equations (4.19), (A.1)–(A.4), and their proofs

For a periodic UU-orbit, translating a fundamental domain removes the smooth-density comparison that fails outside the fully expanding cone, leaving the same thickening and nonescape bounds. In the appendix, Theorem 4.5 applied to the 2m2m time parameters gives exponential two-time correlation decay after centering the last observable. The cited almost-sure criterion then yields the stated T−1/2log⁡3/2+εTT^{-1/2}\log^{3/2+\varepsilon}T error scale, and the induction over the observables is valid in the version of record.

IMRN version of record ↗
02Proofs4 reported findingsContains incorrect or incomplete proofs

The central analytic arguments are complete, but Lemma 3.7 uses a false finite-index power assertion in the step that transfers injectivity radius from a finite-cover quotient. Replacing the subgroup index by its factorial repairs the argument. The remaining defects are local formula, notation, or bibliographic errors.

Lemma 3.7Incomplete as written · verified repair

A subgroup of index nn need not contain every nnth power

Version of record · Section 3.3, Lemma 3.7 · paragraph beginning with the cardinality nn

Let D=ρ(Γ)D=\rho(\Gamma) and E=D∩Γ0E=D\cap\Gamma_0, with [D:E]=n[D:E]=n. The proof asserts ρ(γ)n∈E\rho(\gamma)^n\in E for every γ∈Γ\gamma\in\Gamma. This is false when EE is not normal: for example, an element can act as a transposition on a three-element coset space. Set N=n!N=n! instead. The action of DD on D/ED/E maps every element to a permutation of nn letters, so dNd^N lies in the kernel of that action and hence in EE for every d∈Dd\in D. Replacing nn by NN in the radius bounds and in the bounded-torsion neighborhood gives (g1−1g2)N=1(g_1^{-1}g_2)^N=1 and then g1=g2g_1=g_2, exactly as required. This change affects constants only and proves the claimed injectivity comparison.

IMRN version of record ↗
Section 2.3Typos

The worked SL⁡m+n\operatorname{SL}_{m+n} root data contain two numerical slips

Version of record · Section 2.3, paragraph after the display defining Φ\Phi, Φ+\Phi^+, and Π\Pi

For the upper-right m×nm\times n block unipotent radical, the omitted simple root is em−em+1e_m-e_{m+1}, not em−em−1e_m-e_{m-1}. Also, with the Killing form B(X,Y)=2(m+n)tr⁡(XY)B(X,Y)=2(m+n)\operatorname{tr}(XY) on slm+n\mathfrak{sl}_{m+n}, the diagonal entries of sei−ejs_{e_i-e_j} are 1/(2(m+n))1/(2(m+n)) and −1/(2(m+n))-1/(2(m+n)), not 1/(m+n)1/(m+n) and −1/(m+n)-1/(m+n). The following display correctly lists Φ(u)={ei−em+j}\Phi(\mathfrak u)=\{e_i-e_{m+j}\}, and multiplying every sαs_\alpha by the same positive scalar does not change the cone, so Example 1.1 and Theorem 1.2 remain correct.

IMRN version of record ↗
Theorems 1.3 and 1.5 source chainVerified with exact-form caveats

Unavailable publisher files do not leave the imported claims unsupported

Version of record · Sections 3–4 · citations [2], [7], [11], [16], [18], [20], [23], and [25]

The Cambridge version of record for Buenger–Zheng was not publicly downloadable, but its sole arXiv proof-bearing form was audited in full; the marked rank-one Theorem 1.1 is valid after the explicit constant correction and nilpotent-hull repair recorded there. That fallback's unsupported product-only Theorem 1.5 is not used by Shi. The public typeset Kleinbock–Margulis article was audited, and its unrelated return-time gap does not affect Lemma 2.4.7 used for smoothing. The final public forms of the quantitative-nondivergence and EMV inputs were also audited. The exact older Margulis–Tomanov publisher file was not confirmed open, but its big-cell decomposition used in Lemma 4.1 follows independently from the stable, central, and unstable root-space decomposition. Thus no inaccessible non-book ancestor forces an unsupported-statement or unsupported-source finding for Shi's Theorems 1.3 or 1.5.

IMRN version of record ↗
Lemma 4.3 and reference [32]Typos

One group symbol and one bibliography record are wrong

Version of record · proof of Lemma 4.3 and bibliography entry [32]

The mixing line in Lemma 4.3 writes ⟨φ,g−tψ⟩\langle\varphi,g_{-t}\psi\rangle once, although the one-parameter group is denoted btb_t everywhere; the required term is ⟨φ,b−tψ⟩\langle\varphi,b_{-t}\psi\rangle. Reference [32] identifies the right Shah–Weiss title, authors, theorem number, and equation (3.13), but gives the wrong journal data: the article is in Ergodic Theory and Dynamical Systems 20 (2000), 567–592, not volume 16 (1996), 1–28. The exact public author copy was inspected and confirms the source use, so neither typo changes a proof.

IMRN version of record ↗
03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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