Pith Number

pith:SLAOIEZS

pith:2026:SLAOIEZSTFOPL7IF7XE5DPCYP6

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Determining the Winner in Alternating-Move Games

Auriel Rosenzweig, Itamar Bella\"iche

Hausdorff dimension of the target set determines the winner in two-player alternating-move games on trees.

arxiv:2601.08359 v3 · 2026-01-13 · math.DS · cs.GT · math.LO · math.OC

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Claims

C1strongest claim

We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set.

C2weakest assumption

The target sets admit well-defined Hausdorff dimensions in the metric spaces considered, and the generalized Hausdorff dimension games yield valid lower bounds on those dimensions when Player I has a winning strategy.

C3one line summary

A criterion based on the Hausdorff dimension of the target set determines the winner in alternating-move win-lose games on trees, generalizing prior results on Schmidt games to arbitrary complete metric spaces.

References

19 extracted · 19 resolved · 0 Pith anchors

[1] D. Badziahin and S. Harrap (2017) Cantor-winning sets and their applica- tions, Advances in Mathematics, vol. 318, pp. 627-77 2017

[2] D. Badziahin and S. Harrap and E. Nesharim and D. Simmons (2024) Schmidt games and Cantor winning sets, Ergodic Theory and Dynamical Systems, vol. 45, no. 1, pp. 71-110. https://doi.org/10.1017/etds.2 2024 · doi:10.1017/etds.2024.23

[3] Baek (2009) Dimensionally invariant spaces, Journal of the Chungcheong Mathematical Society, volume 22 no 2009

[4] Bellaïche (2023) Dyadic Hausdorff Dimension Games [Master’s Thesis, Tel Aviv University] 2023

[5] G. David and S. Semmes (1993) Analysis of and on Uniformly Rectifiable Sets, American Mathematical Society 1993

Formal links

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Receipt and verification

First computed	2026-05-18T02:44:31.957456Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

92c0e41332995cf5fd05fdc9d1bc587f8e59e255c2d8f2362e3e3d3ff7518a3e

Aliases

arxiv: 2601.08359 · arxiv_version: 2601.08359v3 · doi: 10.48550/arxiv.2601.08359 · pith_short_12: SLAOIEZSTFOP · pith_short_16: SLAOIEZSTFOPL7IF · pith_short_8: SLAOIEZS

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/SLAOIEZSTFOPL7IF7XE5DPCYP6 \
  | 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: 92c0e41332995cf5fd05fdc9d1bc587f8e59e255c2d8f2362e3e3d3ff7518a3e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e7dc0cda0b6629a8466dd651c3192f6257ed663907f09b53df4e4492ee9ae8f4",
    "cross_cats_sorted": [
      "cs.GT",
      "math.LO",
      "math.OC"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-01-13T09:16:14Z",
    "title_canon_sha256": "7bd785f4127ea955d9f772a676621b4272ab51b1836d6e348afdb993f14be350"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.08359",
    "kind": "arxiv",
    "version": 3
  }
}