arXiv:2605.14574v2

McShane-Rivin norm balls and simple-length multiplicities

Nhat Minh Doan, Xiaobin Li, Van Nguyen

math.GTmath.MGmath.NT30F6057K2052A1011H0611J0611J70

Paper metadata

Original submission: May 14, 2026
Latest revision: July 19, 2026 · version 2
Claimed MSC: 30F60, 57K20, 52A10, 11H06, 11J06, 11J70
arXiv categories: math.GT, math.MG, math.NT
Paper license: CC BY-NC-ND 4.0 ↗
Comments: 32 pages, 5 figures. Comments welcome!

Abstract

We use normal-turn estimates to study the global and local geometry of the boundaries of McShane--Rivin norm balls $B_X$ for complete finite-area hyperbolic once-punctured tori $X$ . This yields a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly $L\geq 2$ is at most $C_X(\log L)^2$ . For the modular torus, this gives $\#λ_M^{-1}(m)\leq C(\log\log(3m))^2$ for every Markoff number $m$ , improving the previous logarithmic bounds for Markoff fibers. Our second result shows that the boundary $\partial B_X$ is a convex-geometric detector of exponential Diophantine approximation: a rational direction gives genuine corner with exponentially small exterior angle in the hyperbolic length of the corresponding simple closed geodesic, while at an irrational direction $β$ the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach $β$ and the $\ell^\infty$ -radius of $B_X$ in the projective direction $β$ . Thus, irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets; in particular, every intermediate finite-flatness level determines the marked torus.

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Audit summary

Audited against arXiv v2

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 15, 2026

01Statements4 reported findingsCorrect

The logarithmic-square lattice and simple-length multiplicity bounds, the rational-corner estimate, the exact irrational flatness formula and its level-set consequences, and the two stated corollaries are correct. The global estimate follows from the proved exponential normal-turn tail and a localized Jarník argument; the local formula follows from matching upper and lower turn estimates at irrational directions.

Theorem 1.1Correct

Logarithmic-square boundary and simple-length counts

Page 4 · Theorem 1.1; proof on pages 17–18 · arXiv:2605.14574v2

For height cutoff $H$ , the boundary points in rational directions of height at most $H$ contribute $O(H^2)$ points. The complementary arcs have total normal turn $O_X(e^{-\kappa_X H})$ and total dilated arclength $O_X(L)$ . Lemma 4.1 and Hölder's inequality therefore give $\#(L\partial B_X\cap\mathbb Z^2)\leq C_XL^{2/3}e^{-\kappa_XH/3}+O(H^2).$ Choosing $H=\lceil 2\kappa_X^{-1}\log L\rceil$ proves the claimed $C_X(\log L)^2$ bound. Restricting to primitive vectors and identifying $w$ with $-w$ gives the simple-length multiplicity estimate.

Full paper, version 2 ↗

Theorem 1.2Correct

Rational corners and exact irrational flatness

Pages 4–5 · Theorem 1.2; proofs on pages 19–29 · arXiv:2605.14574v2

For a primitive class $w$ , the Fricke fan calculation gives a determinant of the endpoint supporting functionals equal in absolute value to $4\|w\|_X\log\coth\!\left(\frac{\|w\|_X}{2}\right),$ which is comparable to $\operatorname{ht}(w)e^{-\|w\|_X}$ ; the matching upper bound follows by placing $w$ in its height- $(\operatorname{ht}(w)-1)$ Farey gap. At an irrational direction the exponential tail makes the local turn tend to zero, so the supporting line is unique. The local turn has lower logarithmic decay rate $\tau_X(\beta)/\Omega(\beta)$ , with the stated conventions at $0$ and $\infty$ . The graph-height estimate supplies the lower flatness bound, while rational corner atoms along a sequence realizing $\Omega(\beta)$ supply the matching obstruction. This yields exactly $\mathfrak f_X(\beta)=1+\frac{\tau_X(\beta)}{\Omega(\beta)}$ in the finite positive case and the printed endpoint values. Continued-fraction constructions and the covering and Baire arguments then give all density, measure, category, and Hausdorff-dimension clauses.

Full paper, version 2 ↗

Corollary 1.4Correct

Finite flatness levels determine the marked torus

Page 6 · Corollary 1.4; proof on pages 29–30 · arXiv:2605.14574v2

On a dense overlap $\mathcal F_A(X)\cap\mathcal F_B(Y)$ , the exact formula gives $\tau_X=c\tau_Y$ with $c=(A-1)/(B-1)$ . Continuity extends this relation to all projective directions, hence $\|\cdot\|_X=c\|\cdot\|_Y$ . McShane's identity rules out $c\neq1$ by strict termwise comparison of the two convergent positive series. Equality of the resulting positive Fricke trace triple for a marked basis and its sum determines the marked cusped torus, giving $A=B$ and $X=Y$ .

McShane, simple-geodesic identity ↗

Corollary 1.5Correct

Markoff-fiber bound

Page 6 · Corollary 1.5 · arXiv:2605.14574v2

On the modular torus, a Markoff number $m$ corresponds to length $L_m=2\operatorname{arcosh}(3m/2)$ . Thus $\log L_m\asymp\log\log(3m)$ outside a bounded initial range, and Theorem 1.1 gives $\#\lambda_M^{-1}(m)\leq C(\log\log(3m))^2.$ Enlarging the constant covers the bounded range, including the smallest Markoff number.

02Proofs5 reported findingsCorrect

The central proof chains are correct and complete. The Fricke recurrences give the required endpoint and corner estimates; Bowditch's sink theorem selects the larger root on every sufficiently high Farey gap; the turn tail sums correctly; and the convex-lattice, continued-fraction, and rigidity arguments preserve all stated constants, regimes, and quantifiers.

Theorem 2.3 and Lemma 2.5Correct and complete

Fricke identities and Bowditch root selection

Pages 8–10 · Theorem 2.3 and Lemma 2.5 · arXiv:2605.14574v2

The positive trace labels satisfy the Vieta relations $x_{u+v}+x_{u-v}=x_ux_v,\qquad x_{u+v}x_{u-v}=x_u^2+x_v^2.$ Since every label is $2\cosh(\|w\|_X/2)>2$ , Bowditch's positive-Fuchsian sink conclusion applies. A height- $H$ Farey gap with $H$ beyond the three sink-label heights lies on the component opposite the sink; hence the arrow on its dual edge points toward $u-v$ , which is exactly the inequality $x_{u+v}\geq x_{u-v}$ used later. The Farey-gap lemma correctly supplies determinant-one endpoint representatives and $\operatorname{ht}(u+v)>H$ .

Bowditch, Markoff triples and quasifuchsian groups ↗

Propositions 3.1–3.2 and Theorem 3.3Correct and complete

Endpoint estimates and exponential normal-turn tail

Pages 13–16 · Section 3 · arXiv:2605.14574v2

Solving the recurrence along $nu+v$ gives $x_{nu+v}=Ae^{na}+O_{u,v}(e^{-na})$ , where $2a=\|u\|_X$ , and therefore $\lambda_{u\mid v}(v)=2\log A$ . The displayed Vieta calculation proves $|2\log A-\|v\|_X|\lesssim_Xe^{-\|u\|_X-\|v\|_X}$ , and symmetrically at the other endpoint. Expressing the functional difference in the determinant-one dual basis incurs only the factor $H$ . The two antipodal gap arcs have equal turn and each has turn at most $\pi$ , so once the endpoint-normal angle is small the relevant swept interval is the smaller one. Summing over $O(H^2)$ gaps and absorbing the polynomial factor into a weaker exponential gives Theorem 3.3 with no omitted case.

Full paper, version 2 ↗

Lemma 4.1 and proof of Theorem 1.1Correct and complete

Localized Jarník count

Pages 16–18 · Section 4 · arXiv:2605.14574v2

Distinct consecutive lattice chords on a strictly convex arc have distinct primitive directions. In an angular sector of width $\theta$ , determinant integrality gives at most $O(\theta r^2+1)$ primitive directions of length at most $r$ . Taking $r\asymp(N/\theta)^{1/2}$ forces at least half of $N$ chord directions to be long, and the arclength budget yields $N\lesssim R^{2/3}\theta^{1/3}+1$ . Splitting a general arc into at most four normal sectors and applying Hölder preserves this estimate. The subsequent sum over the height-gap components uses the total arclength and total turn with the correct exponents.

Full paper, version 2 ↗

Propositions 5.1–5.5 and Theorem 5.7Correct and complete

Corner atoms and the exact flatness exponent

Pages 19–27 · Sections 5.1–5.2 · arXiv:2605.14574v2

The two fan coefficients satisfy $A_+A_-=\coth^2(\|w\|_X/2)$ , so the supporting-functional determinant gives the lower corner bound; the containing Farey gap gives the upper bound. For local turn, $H_\beta(\rho)$ is bounded below by $Q^{-1}\log(1/\rho)$ whenever $Q>\Omega(\beta)$ , while a sequence of rational directions attaining the limsup contributes corner atoms of size at least $\exp(-(\tau_X(\beta)+o(1))\operatorname{ht}(w))$ . These bounds prove the three cases of Proposition 5.5. In graph coordinates, projective distance is comparable to $|x|$ , graph height is controlled by local turn, and a corner of angle $\alpha$ at $x_0$ forces height at least $\alpha|x_0|/2$ at $3x_0/2$ . Applying these estimates in both directions yields the exact supremal flatness order, including $\Omega=0$ and $\Omega=\infty$ .

Full paper, version 2 ↗

Lemmas 5.8–5.10 and Corollary 5.11Correct and complete

Continued fractions, strata, and rigidity

Pages 27–30 · Sections 5.3–5.4 · arXiv:2605.14574v2

For $\beta=[(1,b)]$ , the usual best-approximation inequalities give $\Omega(\beta)=\frac{1}{\max\{1,|b|\}}\limsup_j\frac{\log q_{j+1}}{q_j}.$ Prescribing the next partial quotient realizes every desired value of the resulting flatness order inside any projective interval. The exact-height count $O(h)$ makes every exponentially approximable set have Hausdorff dimension zero, while the open dense limsup construction gives $\{\Omega=\infty\}$ as a dense $G_\delta$ . These facts imply all metric and category statements. The final use of McShane's identity and the positive Fricke parametrization correctly removes the only possible scaling ambiguity between two marked norms.

Full paper, version 2 ↗

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Paper: arXiv:2605.14574v2
Authors listed: Nhat Minh Doan, Xiaobin Li, Van Nguyen
Audit date: August 15, 2026

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McShane-Rivin norm balls and simple-length multiplicities

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Logarithmic-square boundary and simple-length counts

Rational corners and exact irrational flatness

Finite flatness levels determine the marked torus

Markoff-fiber bound

Fricke identities and Bowditch root selection

Endpoint estimates and exponential normal-turn tail

Localized Jarník count

Corner atoms and the exact flatness exponent

Continued fractions, strata, and rigidity

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