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Let $M\\in\\mathbb{Z}^{2\\times2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\\in\\mathbb{Z}$, $B\\geq 2$) and ${\\bf v}\\in\\mathbb{Z}^2$ such that $({\\bf v},M{\\bf v})$ are linearly independent. Then the equation $$MT+\\frac{B-1}{2}{\\bf v} =T\\cup(T+{\\bf v})\\cup (T+2{\\bf v})\\cup \\cdots\\cup(T+(B-2){\\bf v})\\cup(-T) $$ defines a unique nonempty compact set $T$ satisfying $\\overline{T^o}=T$. Moreover, $T$ tiles the plane by the crystallographic group $p2$ generated by the $\\pi$-rotation a"},"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":"1611.04903","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-11-15T15:58:24Z","cross_cats_sorted":[],"title_canon_sha256":"ce41d12b8af64e7df7e07aa416bf749b4de764acee750801d0bbda4ae2dd8c7e","abstract_canon_sha256":"b9a7d28f174c3df8e5587767ad811cad6a30fb593b0b637e82a8f1b3a857b0a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:49.558068Z","signature_b64":"iO1vYYeBRLj+FpQohXH2IYZ5dcXLur7qvAsq0/Bwq2ISv6qtzPlka4gmYLMDoXsBjF/BKYMZMZAblD5v05v9Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e8e204192bca266eeeb3b9d34c68dca5472fa0505ace548d70534bd2c8a6191","last_reissued_at":"2026-05-18T00:58:49.557477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:49.557477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology of a class of $p2$-crystallographic replication tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Beno\\^it Loridant, Shu-Qin Zhang","submitted_at":"2016-11-15T15:58:24Z","abstract_excerpt":"We study the topological properties of a class of planar crystallographic replication tiles. Let $M\\in\\mathbb{Z}^{2\\times2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\\in\\mathbb{Z}$, $B\\geq 2$) and ${\\bf v}\\in\\mathbb{Z}^2$ such that $({\\bf v},M{\\bf v})$ are linearly independent. Then the equation $$MT+\\frac{B-1}{2}{\\bf v} =T\\cup(T+{\\bf v})\\cup (T+2{\\bf v})\\cup \\cdots\\cup(T+(B-2){\\bf v})\\cup(-T) $$ defines a unique nonempty compact set $T$ satisfying $\\overline{T^o}=T$. 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