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Let $Y$ be any set and suppose that $\\Psi : \\mathbb{Z}^N \\rightarrow Y$ is a $\\Gamma$-invariant function. We prove that for every positive integer $m$, there exists a positive integer $k$ with the property that for every finite set $F \\subset \\mathbb{Z}^N$ with $|F| = m$, we have \\[ \\Psi(kF) \\subse"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.01719","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-06T01:15:35Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"5a6775c92cc9e15b2daa6b874b56c873ca4c6c4961e4c08995f03670e34e7398","abstract_canon_sha256":"e04c32c9655f50b9e5f89634fe9e0922c74c8b36451d6dcd3037ce721683b8c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:10.535594Z","signature_b64":"mASFScI9/HlsJGp7DSUUD66HH13ND4SnfTSe93IlChhtD5jMUG5WtYDQxO5+FROyzr/Na2GyTAPW9DHYR7tPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01f3728a850f8d75fb6c4d95be85f181c6292ff3c6f2a2b6701dbc815fd8c80b","last_reissued_at":"2026-05-18T01:25:10.535063Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:10.535063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Twisted patterns in large subsets of $\\mathbb{Z}^N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Kamil Bulinski, Michael Bj\\\"orklund","submitted_at":"2015-12-06T01:15:35Z","abstract_excerpt":"Let $E \\subset \\mathbb{Z}^N$ be a set of positive upper Banach density and let $\\Gamma < \\operatorname{GL}_N(\\mathbb{Z})$ be a finitely generated, strongly irreducible subgroup whose Zariski closure in $\\operatorname{GL}_N(\\mathbb{R})$ is a Zariski connected semisimple group with no compact factors. Let $Y$ be any set and suppose that $\\Psi : \\mathbb{Z}^N \\rightarrow Y$ is a $\\Gamma$-invariant function. We prove that for every positive integer $m$, there exists a positive integer $k$ with the property that for every finite set $F \\subset \\mathbb{Z}^N$ with $|F| = m$, we have \\[ \\Psi(kF) \\subse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01719","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.01719","created_at":"2026-05-18T01:25:10.535141+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.01719v1","created_at":"2026-05-18T01:25:10.535141+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01719","created_at":"2026-05-18T01:25:10.535141+00:00"},{"alias_kind":"pith_short_12","alias_value":"AHZXFCUFB6GX","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"AHZXFCUFB6GXL63M","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"AHZXFCUF","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH","json":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH.json","graph_json":"https://pith.science/api/pith-number/AHZXFCUFB6GXL63MJWK35BPRQH/graph.json","events_json":"https://pith.science/api/pith-number/AHZXFCUFB6GXL63MJWK35BPRQH/events.json","paper":"https://pith.science/paper/AHZXFCUF"},"agent_actions":{"view_html":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH","download_json":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH.json","view_paper":"https://pith.science/paper/AHZXFCUF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.01719&json=true","fetch_graph":"https://pith.science/api/pith-number/AHZXFCUFB6GXL63MJWK35BPRQH/graph.json","fetch_events":"https://pith.science/api/pith-number/AHZXFCUFB6GXL63MJWK35BPRQH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH/action/storage_attestation","attest_author":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH/action/author_attestation","sign_citation":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH/action/citation_signature","submit_replication":"https://pith.science/pith/AHZXFCUFB6GXL63MJWK35BPRQH/action/replication_record"}},"created_at":"2026-05-18T01:25:10.535141+00:00","updated_at":"2026-05-18T01:25:10.535141+00:00"}