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In dimension two, Nivat conjectured that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then $\\eta$ is periodic. Sander and Tijdeman showed that this holds for $k\\leq2$. We generalize their result, showing that Nivat's Conjecture holds for $k\\leq3$. The method involves translating the combinatorial pro"},"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":"1307.0098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-06-29T12:52:35Z","cross_cats_sorted":[],"title_canon_sha256":"32c74bdd319c256a929cc9e67df9e378affb4a821f2461c5a86c66aabae11b3b","abstract_canon_sha256":"647b3ca4a23ec30c65668c822e18b82f4768ad15e256c837ee7e183185eef160"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:34.275433Z","signature_b64":"VgRlqAMrOVoomrljMUnI2jzybK9ZTGr7ptY/o9FNgy1E79hB2BucyxPyIP6ZWVDhxUKAyxYQaUkQgeMDYuW9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8505ea939aaab360c1918b449b107f5dff2a1124476de98500b7237da12ab76e","last_reissued_at":"2026-05-18T03:19:34.274985Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:34.274985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complexity of short rectangles and periodicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Van Cyr","submitted_at":"2013-06-29T12:52:35Z","abstract_excerpt":"The Morse-Hedlund Theorem states that a bi-infinite sequence $\\eta$ in a finite alphabet is periodic if and only if there exists $n\\in\\N$ such that the block complexity function $P_\\eta(n)$ satisfies $P_\\eta(n)\\leq n$. In dimension two, Nivat conjectured that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then $\\eta$ is periodic. Sander and Tijdeman showed that this holds for $k\\leq2$. We generalize their result, showing that Nivat's Conjecture holds for $k\\leq3$. 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