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Counting Markov triples and simple geodesics

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Morten S. Risager

Number Theory · Submitted 26 Oct 2026 · 36 references

Pith's verdict: First o(log x) error term for Markov triple counting, built on a clever geometric lattice-point argument — but the core estimate leans on an unpublished theorem that hasn't been independently verified.

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arXiv ID2610.36069 (version 1)
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CitationMorten S. Risager. "Counting Markov triples and simple geodesics." arXiv:2610.36069 (2026). https://pith.science/paper/F7HGY2IC
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The paper

The paper claims

First sub-logarithmic error for Markov triple counting

The reading

The paper proves that the number of Markov triples with largest entry at most $x$ is $C(\log(3x))^2 + o(\log x)$, the first time the error has been shown to be sub-logarithmic. The previous best error was $O(\log x \log\log x)$. The proof converts the counting problem into counting primitive lattice points in dilations of the McShane–Rivin unit ball on the modular punctured torus. A detailed geometric analysis of this ball, combined with number-theoretic estimates from the Riemann zeta function's zero-free region, yields the improved error. Under the quasi-Riemann hypothesis the error can be further reduced to $O(\log x / (\log\log x)^{1/2})$.

What carries the argument

The key objects are (1) the McShane–Rivin norm $\|\cdot\|_X$ on $\mathbb{R}^2$, whose unit ball $B_X$ has strictly convex boundary with corners at rational directions, and (2) the Doan–Li–Nguyen theorem (Theorem 2) stating that after removing corners coming from primitive lattice points of height at most $H$, the remaining normal turn on $\partial B_X$ is exponentially small in $H$. The arithmetic structure of the modular torus (Lemma 5) gives a quantitative separation between supporting normals and rational directions, enabling a refined lattice-point estimate via a convex-cap diameter bound (Lemma 8). The Möbius inversion step uses zero-free regions of $\zeta(s)$ via the Mertens function to pass from full lattice points to primitive lattice points.

Extended reading notes

Core claim

The central result is Theorem 1: $M(x) = C(\log(3x))^2 + O\big(\log x\, (\log\log\log x)^{1/10} / (\log\log x)^{3/10}\big)$ unconditionally, and with error $O(\log x / (\log\log x)^{1/2})$ under the quasi-Riemann hypothesis. This improves on the previous best error $O(\log x \log\log x)$ due to McShane and Rivin. The constant $C$ is explicit and equals $\operatorname{vol}(B_X)/(12\,\zeta(2)) = 2\operatorname{vol}(B_X)/\pi^2$, where $B_X$ is the McShane–Rivin unit ball for the modular torus. The proof also gives an improved error term for the number of simple closed geodesics on the modular punctured torus (Theorem 13).

Load-bearing premise

The proof depends on the uniform geometric constants in Theorem 2 of Doan, Li, and Nguyen applying to the modular torus with the same exponential decay rate; any hidden dependence on the surface's geometry that worsens those constants would break the final error estimate.

Editorial extensions

If this is right

  • The error term for Markov triple counting is now known to be $o(\log x)$, ending a decade of stagnation at $O(\log x \log\log x)$.
  • If the quasi-Riemann hypothesis is true, the error improves to $O(\log x / (\log\log x)^{1/2})$, which is closer to the conjectured $O_\varepsilon((\log x)^{1/2+\varepsilon})$.
  • The same method yields an improved error for the number of simple closed geodesics on the modular punctured torus: $P_X(R) = (\operatorname{vol}(B_X)/\zeta(2)) R^2 + O(R / (\log R)^{3/10})$ unconditionally.
  • The numerical verification of the Frobenius uniqueness conjecture up to $10^{30000}$ suggests that the main term captures the count accurately, with a difference of only 35 triples at that height.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's approach opens the possibility of extending sub-logarithmic error bounds to other surfaces whose McShane–Rivin ball has arithmetic properties similar to the modular torus, such as congruence covers.
  • The reliance on Doan–Li–Nguyen's theorem means that any uniform improvement of that theorem (e.g., replacing exponential decay by super-exponential decay) would directly sharpen the error in Theorem 1.
  • The numerical data hint that the true error may be as small as $O((\log x)^{1/2})$, which would match the McShane–Rivin conjecture; proving this would likely require fundamentally new ideas beyond the current geometric and analytic toolbox.
Original abstract

We give an asymptotic formula for the number of Markov triples with largest entry below $x$. We show that the error term is $o(\log x)$, which improves upon the previous best known results due to McShane and Rivin. We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-28: 1 request, 3,593 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.97 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We give an asymptotic formula for the number of Markov triples with largest entry below $x$. We show that the error term is $o(\log x)$, which improves upon the previous best known results due to McShane and Rivin. We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We give an asymptotic formula for the number of Markov triples with largest entry below $x$. result 0.63 0.96 0.83 asserted 1.00
s1 We show that the error term is $o(\log x)$, which improves upon the previous best known results due to McShane and Rivin. result 0.96 0.94 0.22 asserted 1.00
s2 We give an explicit error based on classical zero-free regions of the Riemann zeta function, and a sharper estimate under the quasi-Riemann hypothesis. result 0.80 0.93 0.10 asserted 1.00
s3 The proof uses the correspondence between Markov triples and simple closed geodesics on the modular punctured torus and a careful analysis of the McShane-Rivin unit ball. method 1.00 0.81 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Normalized error term The largest numerical verification of the uniqueness conjecture we could find in the literature appears in [28, App. F]: it covers all Markov numbers less than 1015000 (see [29, 4, 2] for earlier experiments). We verified the uniqueness conjecture numerically up to 1030000. The computation took about 90 minutes on a standard laptop. In fact, we have verified the slightly stronger statement that there are no two different ordered Markov triples (p, q, r), (p ′ , q′ , r′ ) with r, r′ ≤ 1030000 satisfying r = r ′ (mod 2128). We found M(1030000) = 862 356 773, w
Figure 2. Fundamental domain for Γ. for which the classes of A and B form the standard basis. The free homotopy classes of oriented, essential, simple, closed curves are parametrized by primitive vectors Z 2 prim = {(p, q) ∈ Z 2 : gcd(p, q) = 1}. Here essential means that the curve is not homotopic to a point or to the puncture/cusp. Each class has a unique simple closed geodesic representative γv [21, Sec. 2]. The geo￾metric intersection number of the curves represented by u and v is | det(u, v)| [9, Sec. 1.2.3]. We call two primitive vectors modulo sign Farey neighbours if | det(u, v)| = 1;
Figure 3. BX for the modular torus where the implied constant is bounded by the length of the boundary of BX. Extra smooth￾ness of a convex region usually results in better error estimates with the Gauss circle problem as the ultimate goal (See e.g. [16, 18]) The error esti￾mation (7) is the basis for the error estimation by McShane and Rivin [21, Thm. 0.1]. For general convex domains (7) is best possible and is attained by the square. In this paper we show that for the McShane￾Rivin ball BX for the once-punctured modular torus we can get a better error term. Understanding the geometry of BX…
Figure 4. λu|v = 1, u = (1, 1), v = (1, 2). By [7, Lem 2.6] we have ∥a + tb∥X − ∥a∥X t → λa|b (b) for t → 0 +. Let µturn be the positive Borel measure on R 2 that measures the angular change in supporting normals of BX. It is supported on ∂BX and normalized by µturn(∂BX) = 2π. See [7] and [30, p. 221 ff.] for additional background on µturn. The measure is denoted by Turn(·) in [7] and by C0(BX, ·) in [30]. For a primitive vector p ∈ Z 2 , we call ∥p∥∞ the height of both p and the corresponding boundary point p/ ∥p∥X . Consider the corners of BX coming from primitive lattice points of
Figure 5. C(θ, δ) of BX, θ = (0, 1). Lemma 8. Let H ≥ 1, and assume (15). Then for every δ > 0 and θ ∈ S 1 , diam(C(θ, δ)) ≪ ® δ/η if θ /∈ S j J η H,j , sH,j + δ/η if θ ∈ J η H,j . Proof. Since CH contains at least eight corners and the total turn is 2π, Theorem 2 and (15) give (2 + ε)η ≤ me−KH ≤ 2π 8 , so 0 < η < π/8. Let e ⊥ θ be a unit vector, and let a± ∈ ∂BX be the unique supporting points with outward unit normals n± = cos η θ ± sin η e. As discussed after (15), Theorem 2(ii) gives normal gaps between consecutive JH,j of length at least me−KH ≥ (2 + ε)η > 2η. The interva

Figures reproduced from arXiv:2610.36069 by the authors.

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