Pith Number

pith:WFZ5RD34

pith:2026:WFZ5RD34GCXLTY6X6XOARADKWI

not attested not anchored not stored refs resolved

${\mathrm{ASL}_n}(\mathbb Z)$ invariant random subsets of $\mathbb Z^n$

Miko{\l}aj Fr\k{a}czyk, Simon Machado

ASL_d(Z)-invariant random subsets of Z^d are built from random equivariant polynomials and independent sampling.

arxiv:2605.16921 v1 · 2026-05-16 · math.PR · math.CO · math.DS

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\usepackage{pith}
\pithnumber{WFZ5RD34GCXLTY6X6XOARADKWI}

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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

We show that every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method.

C2weakest assumption

The proofs rely on the interaction between the Host--Kra theory of characteristic factors, Zimmer's theory of dynamical cocycles of simple Lie groups, and the dynamics of SL_d(Z)-actions on homogeneous spaces (abstract, final paragraph).

C3one line summary

ASL_d(Z)-invariant point processes on Z^d arise from random SL_d(Z)-equivariant polynomials combined with independent site retention.

References

29 extracted · 29 resolved · 0 Pith anchors

[1] Ergodic behavior of diagonal measures and a theorem of Szemer 1977

[2] Kazhdan constants for SL (3, Z). , author=. Journal f. 1991 , volume= 1991

[3] De Finetti, Bruno , booktitle=. La pr

[4] Finite forms of de Finetti's theorem on exchangeability , author=. Synthese , volume=. 1977 , publisher= 1977

[5] The Annals of Probability , pages= 1980

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:30.542185Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b173d88f7c30aeb9e3d7f5dc08806ab2049177b4bd68368b8dc53d12df75799e

Aliases

arxiv: 2605.16921 · arxiv_version: 2605.16921v1 · doi: 10.48550/arxiv.2605.16921 · pith_short_12: WFZ5RD34GCXL · pith_short_16: WFZ5RD34GCXLTY6X · pith_short_8: WFZ5RD34

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WFZ5RD34GCXLTY6X6XOARADKWI \
  | 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: b173d88f7c30aeb9e3d7f5dc08806ab2049177b4bd68368b8dc53d12df75799e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9601fc85cb5f5e1b1b4fdba6bd1cf7674f779c1bd82844a4040eb49455b171d1",
    "cross_cats_sorted": [
      "math.CO",
      "math.DS"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-16T10:27:52Z",
    "title_canon_sha256": "f26e3ddf428337d33560d0a64afb38f8910aae0ef1e3e9186f1c0b6ead2701c0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16921",
    "kind": "arxiv",
    "version": 1
  }
}