{"paper":{"title":"${\\mathrm{ASL}_n}(\\mathbb Z)$ invariant random subsets of $\\mathbb Z^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"ASL_d(Z)-invariant random subsets of Z^d are built from random equivariant polynomials and independent sampling.","cross_cats":["math.CO","math.DS"],"primary_cat":"math.PR","authors_text":"Miko{\\l}aj Fr\\k{a}czyk, Simon Machado","submitted_at":"2026-05-16T10:27:52Z","abstract_excerpt":"We classify measures on $\\{0,1\\}^{\\mathbb{Z}^d}$, $d \\geq 3$, the space of subsets of $\\mathbb{Z}^d$, which are invariant under all affine special linear transformations. In other words, we classify simple point processes on $\\mathbb{Z}^d$ whose law is invariant under affine special linear transformations.\n  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: a random polynomial map is drawn from a distribution invariant under a natural action of $\\mathrm{SL}_d(\\mat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"ASL_d(Z)-invariant point processes on Z^d arise from random SL_d(Z)-equivariant polynomials combined with independent site retention.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"ASL_d(Z)-invariant random subsets of Z^d are built from random equivariant polynomials and independent sampling.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0cae9502402f02bbb8991f5e82874b469a5908b25955438d0886015b9c1b8da6"},"source":{"id":"2605.16921","kind":"arxiv","version":1},"verdict":{"id":"c29c5565-38bb-42ad-85a0-97a6b90288d9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:07:03.462515Z","strongest_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.","one_line_summary":"ASL_d(Z)-invariant point processes on Z^d arise from random SL_d(Z)-equivariant polynomials combined with independent site retention.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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).","pith_extraction_headline":"ASL_d(Z)-invariant random subsets of Z^d are built from random equivariant polynomials and independent sampling."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16921/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T20:22:30.376086Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:18.929630Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:20:47.566984Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.261800Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.342295Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"117da2cbdcaf37e01dc1fa16c0c4627a989751ea8caebf88e884cfc07ee3c89d"},"references":{"count":29,"sample":[{"doi":"","year":1977,"title":"Ergodic behavior of diagonal measures and a theorem of Szemer","work_id":"3fe840b2-037a-4aa4-9249-a0dfe18f7483","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1991,"title":"Kazhdan constants for SL (3, Z). , author=. 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