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Define $\\Gamma(k)$ to be the largest odd $n$ such that the prefix of $\\textbf{t}$ of length $kn$ is not a $k$-antipower, and $\\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $\\gamma(k)$ and $\\Gamma(k)-\\gamma(k)$. Our bounds on $\\gamm"},"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":"1705.06310","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-17T18:49:21Z","cross_cats_sorted":[],"title_canon_sha256":"175559eefecb5912e490903810762af96ea0ffcae527a4e97d8dc5878c030331","abstract_canon_sha256":"398ca9496b9326fe79f7461748c3b9b1c718a7a157178253aba509ee33595469"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:07:43.098651Z","signature_b64":"6nQvjbNHh+4OQuLJsnjfMwB9X5FPCZc36opyX9UTy85BMx+ugw/45TCA8/+GiwLVT9/pz8xKWugwR11lEpVHCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a1a677eb829e4248f89cbaafaee7ed74a9157b03aa8cdb4701510427904b48a","last_reissued_at":"2026-07-05T00:07:43.098232Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:07:43.098232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functions on Antipower Prefix Lengths of the Thue-Morse Word","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shyam Narayanan","submitted_at":"2017-05-17T18:49:21Z","abstract_excerpt":"We say that a word $w$ of length $kn$ is a $k$-\\textit{antipower} if it can be written in the form $w_1 \\cdots w_k$, where each $w_i$ is a distinct word of length $n$. We analyze prefixes of the Thue-Morse word $\\textbf{t}$ and lengths of antipowers occurring in them. Define $\\Gamma(k)$ to be the largest odd $n$ such that the prefix of $\\textbf{t}$ of length $kn$ is not a $k$-antipower, and $\\gamma(k)$ to be the smallest odd $n$ such that the corresponding prefix is a $k$-antipower. We provide strong bounds on the asymptotic values of $\\gamma(k)$ and $\\Gamma(k)-\\gamma(k)$. 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