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arXiv:2001.05174v4

Nondense orbits on homogeneous spaces and applications to geometry and number theory

Jinpeng An, Lifan Guan, Dmitry Kleinbock

math.DS
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Paper metadata

Original submission
January 15, 2020
Latest revision
January 17, 2021 · version 4
Claimed MSC
arXiv categories
math.DS
Comments
Final version. 40 pages

Abstract

Let GG be a Lie group, Γ⊂GΓ\subset G a discrete subgroup, X=G/ΓX=G/Γ, and ff an affine map from XX to itself. We give conditions on a submanifold ZZ of XX guaranteeing that the set of points x∈Xx\in X with ff-trajectories avoiding ZZ is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on XX. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.

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Audit summary

Audited against arXiv v4

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsCorrect

The hyperplane-absolute-winning orbit-avoidance theorems, their geodesic-flow consequences, and the generalized-indefinite-binary-form application are correct as stated.

Theorems A1 and A2Correct

Transversal targets have hyperplane-absolute-winning exceptional sets

Pages 3 and 10–17 · Theorems A1–A2 and Theorems 2.6–2.8 · arXiv:2001.05174v4

The polynomial projection onto the maximally expanding Jordan directions produces an abelian subgroup. Uniform growth outside its kernel reduces each dangerous inverse image of a transversal target to a controlled neighborhood of an affine hyperplane. The hyperplane-percentage strategy deletes a fixed positive fraction of those neighborhoods at every scale, and the equivalence with the hyperplane absolute game gives the stated HAW conclusion. Compact exhaustion and countable-intersection stability cover noncompact targets without changing the hypotheses.

Theorems B1 and B2Correct

Nondense geodesic conclusions on locally symmetric spaces

Pages 4 and 28–31 · Theorems B1–B2 and their proofs · arXiv:2001.05174v4

The homogeneous-space model identifies the relevant sphere of initial directions with a compact-group orbit. The maximally expanding horospherical direction is transverse to each point target, so Theorem A2 gives a thick set of avoiding directions. For a finite subset of an ergodic submanifold, a finite intersection of the corresponding HAW sets is nonempty on every fiber; taking the orbit closure gives a closed invariant set disjoint from the prescribed points and projecting onto the base.

Theorem CCorrect

Value-avoidance for generalized indefinite binary forms

Pages 5 and 31–37 · Theorem C, Theorem 6.3, and Section 7 · arXiv:2001.05174v4

Condition (IB-2) makes a bounded value along an integer vector force the associated lattice orbit into a compactness obstruction, while (IB-3) writes every nonzero level set as a countable union of invariant transverse submanifolds. Theorem A2 applies to each such component, and HAW stability under countable intersections yields avoidance of every value in the prescribed countable set throughout the orbit O(ϕ)\mathcal O(\phi).

02Proofs4 reported findingsCorrect

The Jordan-growth lemmas, the hyperplane-game construction, and both application chains are correct and complete. Two local Jordan-block display slips have uniquely determined repairs and do not affect any argument.

Sections 2–4Correct and complete

Jordan growth and the hyperplane-game proof

Pages 6–27 · Lemmas 2.2–4.7 and proof of Theorem 2.6 · arXiv:2001.05174v4

The polynomial pTp_T kills all submaximal Jordan blocks and extracts the leading coordinate of each maximal block. Lemmas 3.2–3.5 give uniform two-sided growth and convergence to that projection on compact sets away from its kernel. Transversality then gives a uniform angle from the dangerous tangent spaces, and the local multiplication estimates preserve that angle at the game scale. The resulting deletions satisfy the percentage-game rules and exclude every future hit on the target.

Sections 5–7Correct and complete

Geodesic and value-distribution applications

Pages 28–40 · proofs of Theorems B1, B2, and C · arXiv:2001.05174v4

The identifications of tangent directions and lattice orbits preserve the relevant avoidance conditions. The subgroup transversality checks use the direct root-space decompositions, while compact/countable exhaustion is justified by countable stability of HAW sets. The GIBF argument separately handles the zero level through (IB-2) and every nonzero level through (IB-3), so no value or parameter case is omitted.

Equation (2.4)Typo

The number of leading Jordan blocks is labeled by the wrong symbol

Page 7 · Equation (2.4) · arXiv:2001.05174v4

After Equation (2.3), exactly the first r0r_0 Jordan blocks have both maximal spectral modulus and maximal size, but Equation (2.4) labels the repeated E1sE_{1s} blocks as occurring nn times. Replace nn by r0r_0. The kernel and image formulas (2.5)–(2.6) already use r0r_0, so the intended correction is unique and every later argument is unchanged.

Lemma 3.4Typo

The final Jordan-block estimate drops its normalizing binomial factor

Page 14 · final paragraph of the proof of Lemma 3.4 · arXiv:2001.05174v4

For i>r0i>r_0, the exact coefficient formula still contains (ns−1)−1\binom{n}{s-1}^{-1}, but the displayed upper bound omits that factor and the next sentence says ∣λi∣<1|\lambda_i|<1. The ordering in Equation (2.3) also permits ∣λi∣=1|\lambda_i|=1 with si<ss_i<s. Restore the omitted factor in the bound. If ∣λi∣<1|\lambda_i|<1, exponential decay gives the limit; if ∣λi∣=1|\lambda_i|=1, then si<ss_i<s and (nsi−1)/(ns−1)=O(nsi−s)→0\binom{n}{s_i-1}/\binom{n}{s-1}=O(n^{s_i-s})\to0. These are immediate consequences of the preceding exact formula and the stated block ordering, so the correction is unique and the uniform convergence conclusion is unchanged.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2001.05174v4
Authors listed
Jinpeng An, Lifan Guan, Dmitry Kleinbock
Audit date
August 19, 2026
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