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In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\\mathbf{A} \\in \\mathbb{Q}^{n \\times n}$. We define rational self-affine tiles as compact subsets of the open subring $\\mathbb{R}^n\\times \\prod_\\mathfrak{p} K_\\mathfrak{p}$ of the ad\\'ele ring $\\mathbb{A}_K$, where the factor"},"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":"1203.0758","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-04T18:59:13Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f8e8eedf69c60c753374337b29b02d3f3f21afd66360f0e0b8a5aec28dc3faf6","abstract_canon_sha256":"ed72c89b0ee81be2aefa8aff4b48cb2955d6ae81632f8c845c246e4acb703703"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:14:39.410620Z","signature_b64":"hHgtdLRyUpuLSHHlHNz3ljY9FvNo6mHhEA2t03XCDaR0dqRMEcdD+apQl0mOBPIjwV8PrG6iHxNTX4HUMGUmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9257e6e122d56057ff2d3f50a3f9de590c7952595a393115ec3aa7f6f21ae1fd","last_reissued_at":"2026-05-18T03:14:39.410104Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:14:39.410104Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational self-affine tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"J\\\"org Thuswaldner, Wolfgang Steiner (LIAFA)","submitted_at":"2012-03-04T18:59:13Z","abstract_excerpt":"An integral self-affine tile is the solution of a set equation $\\mathbf{A} \\mathcal{T} = \\bigcup_{d \\in \\mathcal{D}} (\\mathcal{T} + d)$, where $\\mathbf{A}$ is an $n \\times n$ integer matrix and $\\mathcal{D}$ is a finite subset of $\\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\\mathbf{A} \\in \\mathbb{Q}^{n \\times n}$. We define rational self-affine tiles as compact subsets of the open subring $\\mathbb{R}^n\\times \\prod_\\mathfrak{p} K_\\mathfrak{p}$ of the ad\\'ele ring $\\mathbb{A}_K$, where the factor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0758","kind":"arxiv","version":2},"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":"1203.0758","created_at":"2026-05-18T03:14:39.410190+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.0758v2","created_at":"2026-05-18T03:14:39.410190+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0758","created_at":"2026-05-18T03:14:39.410190+00:00"},{"alias_kind":"pith_short_12","alias_value":"SJL6NYJC2VQF","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"SJL6NYJC2VQFP7ZN","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"SJL6NYJC","created_at":"2026-05-18T12:27:20.899486+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/SJL6NYJC2VQFP7ZNH5IKH6O6LE","json":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE.json","graph_json":"https://pith.science/api/pith-number/SJL6NYJC2VQFP7ZNH5IKH6O6LE/graph.json","events_json":"https://pith.science/api/pith-number/SJL6NYJC2VQFP7ZNH5IKH6O6LE/events.json","paper":"https://pith.science/paper/SJL6NYJC"},"agent_actions":{"view_html":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE","download_json":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE.json","view_paper":"https://pith.science/paper/SJL6NYJC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.0758&json=true","fetch_graph":"https://pith.science/api/pith-number/SJL6NYJC2VQFP7ZNH5IKH6O6LE/graph.json","fetch_events":"https://pith.science/api/pith-number/SJL6NYJC2VQFP7ZNH5IKH6O6LE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE/action/storage_attestation","attest_author":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE/action/author_attestation","sign_citation":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE/action/citation_signature","submit_replication":"https://pith.science/pith/SJL6NYJC2VQFP7ZNH5IKH6O6LE/action/replication_record"}},"created_at":"2026-05-18T03:14:39.410190+00:00","updated_at":"2026-05-18T03:14:39.410190+00:00"}