Proof-critical dependence graph

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

Statement-restricted proof-dependence graph for the submanifold genericity theorems, the multiplicative Dani correspondence, the full-measure half of the Khintchine-type theorem, and Proposition 1.7. It includes the uncited cusp-volume input required on p. 28, its independent Schmidt 1957 proof repair, and two source-use failures: BEG20 is applied after dropping its strong-spectral-gap hypothesis, and BG23 is cited with the wrong theorem number.

Graph scope33 nodes38 proof-critical linksChecked August 23, 2026

Oriented proof graph

Dependence map

Arrows point from a prerequisite toward the paper whose marked statement uses it.

Solid arrow: headline proof lineage
Dashed arrow: a separately marked side or appendix claim
Dashed square: a terminal book

Hover over, or focus, a square to see its full citation.

Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.

Evidence ledger

Proof-critical links

Quantitative multiple mixing→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineageUnresolved source chain

Focal Theorem 2.3 restates BEG20 Theorem 1.1 and uses it as the mixing hypothesis in abstract Theorem 2.2. This edge is mathematically inapplicable at the focal theorem's printed scope because the focal statement drops BEG's strong-spectral-gap hypothesis.

Citation location: Focal Theorem 2.3 and proof of Theorem 1.3, PDF pp. 9–10; BEG20 Theorem 1.1, arXiv v2 p. 3Verification note: Research status: inapplicable at printed scope. Counterexample: take G=SL_3(R)×SL_3(R), a product of cocompact lattices, M a Cartan segment entirely in factor 1, and a nonconstant smooth φ depending only on factor 2. Every tM-average equals φ, not μ(φ). Repair by adding the full BEG strong-spectral-gap hypothesis or sufficient irreducibility assumptions.

Theorem 1.3 via Theorem 2.3 and Theorem 2.2Claims submanifold genericity and a quantitative rate for the Cartan action on G/Γ for every lattice in a semisimple group whose simple factors have real rank at least two.

Effective multiple equidistribution of translated measures→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineagePartially verified

The general multiple-equidistribution estimate verifies the focal mixing hypothesis for the compact U_{m,n}-orbit and is used again in Lemma 5.4 for the shrinking-annulus test functions.

Citation location: Focal Theorem 2.4, pp. 9–10, and Lemma 5.4, pp. 26–28; BG23 Theorem 1.3, printed pp. 213–214Verification note: Research status: verified with citation correction. The focal pinpoint `[BG23, Theorem 1.1]` is wrong. Theorem 1.1 is the special SL_{m+n}(Z) statement; the applicable general result is Theorem 1.3.

Theorem 1.4 via Theorem 2.4 and Theorem 2.2Gives submanifold genericity for Haar measure on a compact U_{m,n}-orbit in SL_{m+n}(R)/Γ.
Theorem 1.9, full-measure halfUses BG23 multiple equidistribution for shrinking cusp annuli and requires μ(Ω_t) ≍ exp(-(m+n)R(t)).

Central limit theorems for group actions which are exponentially mixing of all orders→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineageVerified

The focal configuration-space exhaustion is a Euclidean specialization of BG20 Proposition 6.2 and supplies the partition used in the h-th moment estimate.

Citation location: Focal Lemma 3.4 and proof, pp. 12–15; BG20 Proposition 6.2, arXiv v1 p. 12 and §10

Lemma 3.4 (exhausting configuration space)Partitions h-tuples into clustered and well-separated regions for the moment estimate underlying Theorems 2.1–2.2.

On multiplicatively badly approximable vectors→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineageVerified

The focal paper directly imports the ψ↔R construction and multiplicative Dani correspondence as Lemma 4.1 and Proposition 4.2.

Citation location: Focal pp. 17–19; FK arXiv:2211.04523v3 Lemma 4.1 and Proposition 4.4, pp. 13–15Verification note: Historical version choice is v3; v4 postdates focal v1.

Lemma 4.1 and Proposition 4.2Constructs R from ψ and gives the finite-scale multiplicative Dani correspondence used in Corollaries 4.3–4.5 and Theorems 1.8–1.9.

Mean values over the space of lattices→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineagePartially verified

Rogers' primitive moment formulas provide the uncited cusp-volume estimate. For D=m+n and B_r=(-r,r)^D, they give μ{δ<r}=2^{D-1}ζ(D)^{-1}r^D+O_D(r^{2D}); subtracting radii e^{-R(t)} and e^{-R(t)-1} yields μ(Ω_t) ≍ e^{-DR(t)}.

Citation location: Missing at focal p. 28, where Lemma 5.3 is incorrectly credited; Rogers Theorems 4–5, printed pp. 251–253 and 279–284Verification note: Rogers supplies the exact statement used by the focal claim, but the VOR proof is defective. The Schmidt 1957 direct edge is required to make the proof chain valid.

Theorem 1.9, full-measure halfUses BG23 multiple equidistribution for shrinking cusp annuli and requires μ(Ω_t) ≍ exp(-(m+n)R(t)).

Mittelwerte über Gitter→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineageVerified

The focal page-28 cusp-annulus estimate needs primitive first and second moments for the sup-norm cube. Rogers states the required formula, but his printed proof inherits an invalid transfer step. Schmidt's independent arbitrary-Borel formula repairs that proof input; combined with the primitive Möbius decomposition, it yields μ{δ<r}=2^{D-1}ζ(D)^{-1}r^D+O_D(r^{2D}) for D=m+n≥3 and hence μ(Ω_t)≍e^{-DR(t)}.

Citation location: Uncited at focal p. 28; Schmidt 1957 Satz 2–3 and Lemma 4, printed pp. 273–276, together with the primitive decomposition displayed in Rogers 1955, pp. 279–282Verification note: This is a direct repair-source edge to the focal claim, not a historical citation edge to Rogers.

Theorem 1.9, full-measure halfUses BG23 multiple equidistribution for shrinking cusp annuli and requires μ(Ω_t) ≍ exp(-(m+n)R(t)).

Geometrie der Zahlen→Submanifold-genericity of $\mathbb{R}^d$ -actions and uniform multiplicative Diophantine approximationHeadline lineageTerminal source

The convex-body theorem produces the nonzero lattice point in Ξ(Y,T,c), after which the focal AM–GM estimates are internal.

Citation location: Focal proof of Proposition 1.7, p. 30

Proposition 1.7Universal multiplicative Dirichlet bound obtained through a symmetric convex body and Minkowski's theorem.

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity→Quantitative multiple mixingHeadline lineagePartially verified

BEG cites Katok–Spatzier for passing from K-finite coefficient decay to exponential Sobolev decay for smooth vectors, the two-mixing base estimate of its higher-order induction.

Citation location: BEG arXiv v2 pp. 2–3, equation (1.3), citation [20]; Katok–Spatzier Theorem 3.1 and Corollary 3.2, printed pp. 140–142Verification note: Research status: source scope gap. Katok–Spatzier Corollary 3.2 only treats irreducible cocompact quotients and excludes SO(n,1)/SU(n,1), so that pinpoint alone does not justify BEG's full strong-spectral-gap setting. BEG also cites the broader matrix-coefficient literature, but gives no single fully matching source pinpoint here.

Theorem 1.1Exponential mixing of every order for a semisimple action with strong spectral gap.

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces→Quantitative multiple mixingHeadline lineageVerified

BEG adopts EMV's weighted Sobolev norm construction and the embedding, action-growth, and product estimates used in the coupling induction.

Citation location: BEG §2.2, pp. 11–12, citation [10]; EMV Sobolev sections

Sobolev norm system (2.10)–(2.14)Weighted Sobolev norms with embedding, action-growth, and product estimates used throughout the coupling induction.

Expanding cone and applications to homogeneous dynamics→Effective multiple equidistribution of translated measuresHeadline lineageVerified

Shi's effective k=1 equidistribution of a translated compact U-orbit verifies BG23's EQ1 hypothesis; BG23's abstract theorem bootstraps that one-point estimate to every order.

Citation location: BG23 printed p. 215, statement that EQ1 was established in [11]; Shi Theorem 1.5 specialized to k=1, p. 6 and proof pp. 25–27

Theorem 1.3Effective multiple equidistribution for Wiener measures on compact abelian unipotent orbits in the general semisimple/parabolic setting.

Bounded orbits of nonquasiunipotent flows on homogeneous spaces→Effective multiple equidistribution of translated measuresHeadline lineageVerified

Kleinbock–Margulis Corollary 2.4.4 supplies BG23's EQ2 exponential mixing estimate for the ambient invariant measure.

Citation location: BG23 printed p. 215, citation [6, Corollary 2.4.4]; KM96 author PDF printed p. 13

Theorem 1.3Effective multiple equidistribution for Wiener measures on compact abelian unipotent orbits in the general semisimple/parabolic setting.

Logarithm laws for flows on homogeneous spaces→On multiplicatively badly approximable vectorsHeadline lineageSelf-contained leaf

KM99 Lemma 8.3 is the exact change-of-variables result underlying FK Lemma 4.1; FK Proposition 4.4 then proves the multiplicative correspondence internally.

Citation location: FK v3 Lemma 4.1, p. 13, citation [27, Lemma 8.3]; KM99 Lemma 8.3, pp. 23–24

Lemma 4.1 and Proposition 4.4The change-of-variables function R and multiplicative Dani correspondence copied into the focal paper.

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity→Bounded orbits of nonquasiunipotent flows on homogeneous spacesHeadline lineageVerified

KM96 Theorem 2.4.3 begins with Katok–Spatzier's smooth-vector estimate and extends it from the restricted cocompact setting to representation families separated from the trivial representation.

Citation location: KM96 §2.4.3, printed p. 13; Katok–Spatzier Theorem 3.1

Corollary 2.4.4Uniform exponential matrix-coefficient decay for a family of representations isolated from the trivial representation on every simple factor; this is BG23's EQ2 input.

Sur les coefficients des représentations unitaires des groupes de Lie simples→Bounded orbits of nonquasiunipotent flows on homogeneous spacesHeadline lineagePartially verified

KM96 explicitly uses Cowling §3.1 to obtain a uniform strongly-L^p exponent for irreducible representations outside a fixed neighborhood of the trivial representation, removing Katok–Spatzier's factor exclusions.

Citation location: KM96 §2.4.3, printed p. 13, citation [Cow, §3.1]Verification note: The use and locator are exact in KM96; Cowling's closed primary proof remains inaccessible.

Corollary 2.4.4Uniform exponential matrix-coefficient decay for a family of representations isolated from the trivial representation on every simple factor; this is BG23's EQ2 input.

Representations of Nilpotent Lie Groups and Their Applications, Part I→Expanding cone and applications to homogeneous dynamicsHeadline lineageTerminal source

Exponential coordinates identify U with its Lie algebra and Haar measure with Lebesgue measure, enabling polynomial-good and polar-coordinate estimates.

Citation location: Shi §3 opening, p. 16, citation [8, Theorem 1.2.10(a)]

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

On effective equidistribution of expanding translates of certain orbits in the space of lattices→Expanding cone and applications to homogeneous dynamicsHeadline lineageSelf-contained leaf

Proposition 3.5 turns Shi's shortest-vector compactum estimate into the injectivity-radius form of quantitative nonescape.

Citation location: Shi p. 18 before (3.5), citation [20, Proposition 3.5]; source p. 8

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

Khintchine-type theorems on manifolds: The convergence case for standard and multiplicative versions→Expanding cone and applications to homogeneous dynamicsHeadline lineageVerified

The polynomial-good lemma verifies uniform goodness of the exterior-power covolume functions in Shi's arithmetic nondivergence branch.

Citation location: Shi p. 18, citation [2, Lemma 3.2]; BKM source p. 8

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

An extension of quantitative nondivergence and applications to Diophantine exponents→Expanding cone and applications to homogeneous dynamicsHeadline lineageSelf-contained leaf

Theorem 2.2 turns Shi's verified good-function and lower-bound hypotheses into the O(ε^δ) bad-set estimate (3.5).

Citation location: Shi p. 18, citation [16, Theorem 2.2]; source pp. 9–10

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

Modified Schmidt games and a conjecture of Margulis→Expanding cone and applications to homogeneous dynamicsHeadline lineageVerified

Proposition 3.1 provides finite cusp vectors, precompactness, and short-vector uniqueness, quoted as Shi Lemma 3.4.

Citation location: Shi Lemmas 3.4–3.5, p. 19, citation [23, Proposition 3.1]; source pp. 7–9

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

Non-divergence of unipotent flows on quotients of rank-one semisimple groups→Expanding cone and applications to homogeneous dynamicsHeadline lineageVerified

The explicit rank-one nondivergence constant controls one-dimensional slices and is integrated in polar coordinates in Shi (3.10).

Citation location: Shi p. 20, equation (3.10), citation [7, Theorem 1.1]; source p. 2

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

Discrete Subgroups of Lie Groups→Expanding cone and applications to homogeneous dynamicsHeadline lineageTerminal source

The Borel-density consequence that Z_GΓ is discrete permits Shi's finite-center reduction before splitting into arithmetic and rank-one factors.

Citation location: Shi Lemma 3.6 proof, p. 21, citation [27, Corollary 5.17]

Theorem 1.3Quantitative nonescape controlling the bad slice set in Shi's k=1 proof.

Invariant measures for actions of unipotent groups over local fields on homogeneous spaces→Expanding cone and applications to homogeneous dynamicsHeadline lineageSelf-contained leaf

The stable-central-unstable product theorem gives Shi's open conull cell and Haar integration formula (4.1).

Citation location: Shi Lemma 4.1, p. 22, citation [25, Proposition 2.7]

Lemma 4.1 and formula (4.1)Stable-central-unstable conull cell and Haar decomposition for thickening.

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces→Expanding cone and applications to homogeneous dynamicsHeadline lineageVerified

EMV converts spectral gap into a temperedness exponent and product-group Sobolev coefficient bound, giving Shi's exponential decay along b_t.

Citation location: Shi Lemma 4.2, p. 23, citation [11, §6.2.2]; EMV §§6.2.3, 6.3.2–6.3.3 and Appendix C

Lemma 4.2Converts spectral gap to exponential Sobolev matrix-coefficient decay along the chosen diagonal element.

Bounded orbits of nonquasiunipotent flows on homogeneous spaces→Expanding cone and applications to homogeneous dynamicsHeadline lineageSelf-contained leaf

KM96 Lemma 2.4.7 supplies normalized smooth bumps and Sobolev tensor/product control used in Shi's thickening and estimate (4.18).

Citation location: Shi Lemma 4.4, p. 24, and bump before (4.10), p. 26, citation [18, Lemma 2.4.7]; KM96 printed p. 14

Lemmas 4.3–4.4Controlled bump-function thickening and its Sobolev loss.

Flows on homogeneous spaces and Diophantine approximation on manifolds→Khintchine-type theorems on manifolds: The convergence case for standard and multiplicative versionsHeadline lineageSelf-contained leaf

The one-variable polynomial estimate and multivariable derivative induction prove BKM's polynomial-good Lemma 3.2.

Citation location: BKM Lemma 3.2 proof, p. 8, citing KM98 Proposition 3.2 and Lemma 3.3; source pp. 6–9

Lemma 3.2Polynomial-good estimate used for Shi's exterior-power covolume functions.

Fundamental domains for lattices in $(\mathbb{R}$ -)rank 1 semisimple Lie groups→Modified Schmidt games and a conjecture of MargulisHeadline lineageSelf-contained leaf

Finite-cusp reduction, cusp lattices, and overlap separation establish the compactness and uniqueness parts of Kleinbock–Weiss Proposition 3.1.

Citation location: Kleinbock–Weiss pp. 8–9, citation [GR, Theorem 0.6]

Proposition 3.1Finite cusp-vector family with discreteness, precompactness, and uniqueness of a sufficiently short vector.

Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory→Modified Schmidt games and a conjecture of MargulisHeadline lineageTerminal source

The Dani–Margulis discreteness theorem supplies part (1) of the cusp-vector proposition.

Citation location: Kleinbock–Weiss p. 8, citation [KSS, Theorem 3.4.11]

Proposition 3.1Finite cusp-vector family with discreteness, precompactness, and uniqueness of a sufficiently short vector.

Discrete Subgroups of Lie Groups→Non-divergence of unipotent flows on quotients of rank-one semisimple groupsHeadline lineageTerminal source

Rank-one Bruhat decomposition and a Zassenhaus neighborhood support the classification and linearization of small discrete nilpotent pieces.

Citation location: Buenger–Zheng §2 p. 3 and Lemmas 3.1–3.3, citations to §12.14 and Theorem 8.16

Theorem 1.1Explicit rank-one unipotent nondivergence with the dependence Shi integrates in polar coordinates.

Quantitative nondivergence and its Diophantine applications→Non-divergence of unipotent flows on quotients of rank-one semisimple groupsHeadline lineageTerminal source

Good-function and arbitrary-discrete-lattice quantitative nondivergence yield Buenger–Zheng Theorem 4.1 and Corollary 4.1.

Citation location: Buenger–Zheng §§4.2–4.3, citations [K10, Theorems 3.4 and 3.9]

Theorem 1.1Explicit rank-one unipotent nondivergence with the dependence Shi integrates in polar coordinates.

Almost $L^2$ matrix coefficients→Effective equidistribution for closed orbits of semisimple groups on homogeneous spacesHeadline lineageSelf-contained leaf

The L^{2+ε} characterization and Harish-Chandra majorization convert temperedness into EMV's explicit coefficient estimates.

Citation location: EMV §§6.2.2 and 6.3.2, citation [9, Theorem 1] and equation (6.7)

(6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

Sur les coefficients des représentations unitaires des groupes de Lie simples→Effective equidistribution for closed orbits of semisimple groups on homogeneous spacesHeadline lineagePartially verified

Cowling's integrability theorems give uniform finite L^p exponents for higher-rank and property-(T) rank-one factors.

Citation location: EMV Appendix C, p. 80, citations [8, Theorems 2.4.2 and 2.5.2]Verification note: The statements and numbers are corroborated; the primary proof chain is inaccessible.

(6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

Representation Theory of Semisimple Groups: An Overview Based on Examples→Effective equidistribution for closed orbits of semisimple groups on homogeneous spacesHeadline lineageTerminal source

Spherical-function bounds and Langlands-quotient asymptotics close EMV's remaining rank-one estimates.

Citation location: EMV (6.3) and Appendix C pp. 80–81, citing Proposition 7.15(c), Theorem 8.32, Proposition 8.61

(6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

Intertwining operators for semisimple groups→Effective equidistribution for closed orbits of semisimple groups on homogeneous spacesHeadline lineageSelf-contained leaf

Critical-abscissa and endpoint-unitarity results make nontrivial complementary-series endpoints discrete below 1.

Citation location: EMV Appendix C, p. 80, citation [31, Theorem 6 and Proposition 45]

(6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

Asymptotic properties of unitary representations→Effective equidistribution for closed orbits of semisimple groups on homogeneous spacesHeadline lineageSelf-contained leaf

Vanishing at infinity rules out a nontrivial unitary Langlands quotient at endpoint z=1.

Citation location: EMV Appendix C, pp. 80–81

(6.1), (6.3), (6.9), and (6.10)Converts spectral gap to a uniform temperedness exponent and Sobolev matrix-coefficient decay, including product groups.

On a notion of rank for unitary representations of the classical groups→First cohomology of Anosov actions of higher rank abelian groups and applications to rigidityHeadline lineagePartially verified

Howe's K-finite coefficient estimate is the starting inequality in Katok–Spatzier Theorem 3.1.

Citation location: Katok–Spatzier printed p. 140, citation [5, Corollary 7.2 and §7]Verification note: The exact source chapter is closed; the locator and role are explicit in the inspected KS VOR.

Theorem 3.1 and Corollary 3.2Passes from K-finite matrix-coefficient estimates to Sobolev decay for smooth vectors in its restricted cocompact setting.

Sur les coefficients des représentations unitaires des groupes de Lie simples→First cohomology of Anosov actions of higher rank abelian groups and applications to rigidityHeadline lineagePartially verified

Cowling supplies strong-L^p integrability, uniform in the representation under Katok–Spatzier's excluded-factor hypothesis.

Citation location: Katok–Spatzier printed pp. 140–142, citation [2]

Theorem 3.1 and Corollary 3.2Passes from K-finite matrix-coefficient estimates to Sobolev decay for smooth vectors in its restricted cocompact setting.

Harmonic Analysis on Semi-Simple Lie Groups I→First cohomology of Anosov actions of higher rank abelian groups and applications to rigidityHeadline lineageTerminal source

Warner's K-type eigenvalue and dimension estimates make the Sobolev-weighted isotypic sums converge in the proof of Theorem 3.1.

Citation location: Katok–Spatzier printed p. 141, citations [22, Lemmas 4.4.2.2–4.4.2.3]

Theorem 3.1 and Corollary 3.2Passes from K-finite matrix-coefficient estimates to Sobolev decay for smooth vectors in its restricted cocompact setting.

Ergodic Theory and Semisimple Groups→First cohomology of Anosov actions of higher rank abelian groups and applications to rigidityHeadline lineageTerminal source

Moore ergodicity excludes invariant vectors under noncompact elements and gives discrete kernels for nontrivial irreducible components.

Citation location: Katok–Spatzier Corollary 3.2 proof, printed p. 141, citation [23]

Theorem 3.1 and Corollary 3.2Passes from K-finite matrix-coefficient estimates to Sobolev decay for smooth vectors in its restricted cocompact setting.