MathAudit

Paper metadata

Original submission
January 21, 2012
Latest revision
April 11, 2012 · version 2
Claimed MSC
arXiv categories
math.MG, math.DS, math.NT
Comments
13 pages
Journal reference
GEOMETRIAE DEDICATA 163, ISSUE 1 (2013), 127--140

Abstract

For a proper, geodesic, Gromov hyperbolic metric space XX, a discrete subgroup of isometries Γ\Gamma whose limit set is uniformly perfect, and a disjoint collection of horoballs {Hj}\{H_j\}, we show that the set of limit points badly approximable by {Hj}\{H_j\} is absolutely winning in the limit set. As an application, we deduce that for a geometrically finite Kleinian group acting on Hn+1\mathbb{H}^{n+1}, the limit points badly approximable by parabolics is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that the set of badly approximable limit points has dimension equal to the critical exponent of the group. Since this set can alternatively be described as the limit points representing bounded geodesics in the quotient Hn+1/Γ\mathbb{H}^{n+1}/\Gamma, we recapture a result originally due to Bishop and Jones.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 September 26, 2026
01Statements2 reported findingsNo statement issues found

The absolute-winning theorem for boundary points badly approximable by disjoint horoballs, its Kleinian-group consequences, and the critical-exponent dimension conclusion are correct.

Theorems 1.1 and 3.1No issue found

Badly approximable boundary points are absolute winning

Pages 2 and 7–8 · Theorems 1.1 and 3.1 · arXiv:1201.4505v2

For Bob's ball B(xi,ρi)B(x_i,\rho_i), the strategy deletes a βρi\beta\rho_i-ball about the base point of the unique horoball entered by the ray γxi\gamma_{x_i} at time −log⁡a(ρi)-\log_a(\rho_i). Tree approximation and quasiconvexity show that whenever a surviving ray enters a fixed uniform contraction of a horoball, one of these deletions occurs at a scale comparable to that horoball's shadow radius. Avoiding the deletion therefore gives a uniform positive lower bound for ρ(x,ξj)/Rj\rho(x,\xi_j)/R_j. Uniform perfectness makes the limit set diffuse, so the ambient strategy restricts to Λ(Γ)\Lambda(\Gamma) as claimed.

Exact arXiv version 2 ↗
Corollaries 4.3–5.3No issue found

The Kleinian and Hausdorff-dimension consequences follow

Pages 9–11 · Corollaries 4.3, 4.4, 4.5, and 5.3 · arXiv:1201.4505v2

For a geometrically finite Kleinian group, standard invariant cusp horoballs are disjoint and the limit set is uniformly perfect, so the main theorem applies to BA(Γ)\mathrm{BA}(\Gamma). The bounded-ray interpretation is the usual horoball-excursion criterion, and the two endpoints of a bounded bi-infinite geodesic give the product statement off the diagonal. For the dimension lower bound, Fishman–Simmons–Urbański later proved that for every 0<s<δ0<s<\delta the uniformly radial limit set contains an Ahlfors ss-regular subset; their Remark 1.12 explicitly identifies the weaker statement used here as Mayeda–Merrill Lemma 5.2. Winning on those supports and letting s↑δs\uparrow\delta gives the lower bound, while containment in the conical limit set gives the upper bound.

Fishman–Simmons–Urbański, Theorem 5.9 and Remark 1.12 ↗
02Proofs9 reported findingsNo proof issues found

The shadow-avoidance strategy, its restriction to the limit set, and the Kleinian and dimension applications are complete. The reviewed text has several uniquely repairable notation, constant, and cross-reference defects, none of which changes a theorem or its proof mechanism.

Theorem 3.1No issue found in the reviewed proof

The absolute-game strategy controls every horoball at the correct scale

Pages 7–8 · proof of Theorem 3.1 and Lemma 3.2 · arXiv:1201.4505v2

The four-ray tree approximation gives d(γx(ti),γxi(ti))≤4δd(\gamma_x(t_i),\gamma_{x_i}(t_i))\leq4\delta. The depth interval of the contracted horoball is longer than one legal time increment, so some sampled ray enters the original horoball. The visual-metric shadow estimate then yields cRH≤ρicR_H\leq\rho_i, and survival of Alice's deletion yields ρ(x,ξH)>βρi\rho(x,\xi_H)>\beta\rho_i. These inequalities give a positive horoball-independent approximation constant. Horoballs whose contracted part is met only in the initial bounded ray segment form a finite exceptional family at that fixed contraction depth and contribute a positive finite minimum; a ray aimed at their base point instead meets the contracted horoball arbitrarily late. Thus the lemma's t>t1t>t_1 scope does not leave an uncontrolled family.

Exact arXiv version 2 ↗
Theorem 3.1Suspected typo

A non-singleton absolute-game outcome uses the same argument

Page 7 · first paragraph of the proof of Theorem 3.1 · arXiv:1201.4505v2

The proof writes ⋂iBi={x}\bigcap_i B_i=\{x\} although the definition on page 4 correctly notes that the intersection need not be a point. Replace this by `choose x∈⋂iBix\in\bigcap_i B_i'. Properness makes the visual boundary compact, so the nested closed balls have nonempty intersection, and every subsequent estimate uses only x∈Bix\in B_i, not uniqueness. The proof therefore applies unchanged to any chosen outcome point.

Lemma 3.2Suspected minor formal issue

The shadow estimate is proved for a containing radius, not literally for the diameter

Pages 7–8 · statement and final estimate of Lemma 3.2 · arXiv:1201.4505v2

The statement prints c diam⁡(Sh⁡(H))≤ρic\,\operatorname{diam}(\operatorname{Sh}(H))\leq\rho_i, whereas the proof derives cRH≤ρicR_H\leq\rho_i for a radius satisfying Sh⁡(H)⊆B(ξH,RH)\operatorname{Sh}(H)\subseteq B(\xi_H,R_H). Since diam⁡(Sh⁡(H))≤2RH\operatorname{diam}(\operatorname{Sh}(H))\leq2R_H, replace the displayed cc in the diameter formulation by c/2c/2, or state the radius inequality actually proved. The following paragraph uses ρi/RH\rho_i/R_H, so the downstream argument already uses the valid form.

Lemma 2.4Suspected minor formal issue

The diffuseness lemma is metric, despite its Euclidean wording

Page 5 · Lemma 2.4 and its use on page 8 · arXiv:1201.4505v2

Lemma 2.4 is stated for E⊆RnE\subseteq\mathbb R^n but is later applied to the visual-metric limit set Λ(Γ)⊆∂X\Lambda(\Gamma)\subseteq\partial X. Replace Rn\mathbb R^n by an arbitrary metric space. The printed proof uses only the triangle inequality and the annulus supplied by uniform perfectness, so it proves this corrected scope verbatim and supplies the exact diffuseness input required by Theorem 2.1.

Section 4Suspected typo

The curvature-−1-1 ball metric is missing its factor of two

Page 8 · opening paragraph of Section 4 · arXiv:1201.4505v2

The paper prints ds=∣dx∣/(1−∣x∣2)ds=|dx|/(1-|x|^2) while describing the ball model of sectional curvature −1-1. The standard normalization is ds=2∣dx∣/(1−∣x∣2)ds=2|dx|/(1-|x|^2); the printed normalization has curvature −4-4. The proof uses only the hyperbolic geodesics, horoballs, and coarse boundedness properties, which are unchanged by this constant rescaling.

Corollary 4.5Suspected typo

The diffuseness citation has the wrong lemma number

Page 10 · proof of Corollary 4.5 · arXiv:1201.4505v2

The proof cites nonexistent Lemma 3.3 to choose z∈Λ(Γ)z\in\Lambda(\Gamma) with B(z,βρ)⊆B(y,ρ)∖B(x,βρ)B(z,\beta\rho)\subseteq B(y,\rho)\setminus B(x,\beta\rho). The intended citation is Lemma 2.4, whose conclusion is exactly that Λ(Γ)\Lambda(\Gamma) is diffuse and supplies this ball.

Corollary 4.5Suspected typo

The endpoint identification uses the unit tangent sphere

Page 10 · first paragraph of the proof of Corollary 4.5 · arXiv:1201.4505v2

The displayed identification ToHn+1≅∂Hn+1T_o\mathbb H^{n+1}\cong\partial\mathbb H^{n+1} should read To1Hn+1≅∂Hn+1T_o^1\mathbb H^{n+1}\cong\partial\mathbb H^{n+1}, where To1T_o^1 is the unit tangent sphere. The following map u↦u+u\mapsto u^+ and all later endpoint arguments already require unit tangent vectors, so the intended correction is unique.

Lemma 5.1Suspected typo

The power-law exponents have the wrong sign

Page 11 · statement of Lemma 5.1 · arXiv:1201.4505v2

The paper prints C−1ρ−δ≤μ(B(x,ρ))≤Cρ−δC^{-1}\rho^{-\delta}\leq\mu(B(x,\rho))\leq C\rho^{-\delta}. A locally finite Ahlfors power law has positive exponent: C−1ρδ≤μ(B(x,ρ))≤CρδC^{-1}\rho^{\delta}\leq\mu(B(x,\rho))\leq C\rho^{\delta}. Lemma 5.1's dimension conclusion and the application to the exponent δ−ε\delta-\varepsilon use precisely the positive-exponent form, so this is a sign typo only.

Lemma 5.2 and Corollary 5.3No issue found in the reviewed proof

The communicated Ahlfors-regular subset input is independently verified

Page 11 · Lemma 5.2 and proof of Corollary 5.3 · arXiv:1201.4505v2

The paper cites the construction as a private communication. Fishman–Simmons–Urbański subsequently proved in Theorem 5.9 of arXiv:1301.5630 that for every 0<s<δ0<s<\delta there is an Ahlfors ss-regular subset of the uniformly radial limit set, and Remark 1.12 expressly says that its weaker form was cited as Mayeda–Merrill Lemma 5.2. Positive-dimensional Ahlfors regularity gives uniform perfectness, hence diffuseness, so the winning-set dimension argument in Corollary 5.3 is fully supported.

Fishman–Simmons–Urbański, arXiv:1301.5630 ↗
03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:1201.4505v2
Authors listed
Dustin Mayeda, Keith Merrill
Audit date
September 26, 2026
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