Pith Number

pith:TTPLCJB7

pith:2026:TTPLCJB725V775Z4IBQRTSN2SR

not attested not anchored not stored refs resolved

McShane-Rivin norm balls and simple-length multiplicities

Nhat Minh Doan, Van Nguyen, Xiaobin Li

For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.

arxiv:2605.14574 v1 · 2026-05-14 · math.GT · math.MG · math.NT

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\usepackage{pith}
\pithnumber{TTPLCJB725V775Z4IBQRTSN2SR}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

For every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of length exactly L ≥ 2 is at most C_X (log L)^2. For the modular torus this yields #λ_M^{-1}(m) ≤ C (log log(3m))^2.

C2weakest assumption

The normal-turn estimates for McShane-Rivin norm balls hold uniformly enough to convert growth control into a multiplicity bound without additional length-dependent losses.

C3one line summary

On hyperbolic once-punctured tori the multiplicity of any simple geodesic length L is at most C (log L)^2, improving prior logarithmic bounds for Markoff numbers.

References

36 extracted · 36 resolved · 0 Pith anchors

[1] 2013 , doi = 2013

[2] Khinchin, Aleksandr Yakovlevich , title =

[3] Rockett, Andrew M. and Sz. Continued Fractions , publisher =

[4] Durrett, Rick , title =

[5] Cassels, J. W. S. , title =

Receipt and verification

First computed	2026-05-17T23:39:05.436047Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5

Aliases

arxiv: 2605.14574 · arxiv_version: 2605.14574v1 · doi: 10.48550/arxiv.2605.14574 · pith_short_12: TTPLCJB725V7 · pith_short_16: TTPLCJB725V775Z4 · pith_short_8: TTPLCJB7

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TTPLCJB725V775Z4IBQRTSN2SR \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 9cdeb1243fd76bfff73c406119c9ba945c9f8182d254cf88852dbfb530d0fba5

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "655657a2975d4c837c852cd9ede09737f16f0a0901182948ef98ba5e423ee8b9",
    "cross_cats_sorted": [
      "math.MG",
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-05-14T08:45:18Z",
    "title_canon_sha256": "63f88bbdaf08e38eb7b8a14097c69ec944fa97fac64aecbda211f058a3b93205"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14574",
    "kind": "arxiv",
    "version": 1
  }
}