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These are precisely the values of $n \\geq 3$ with $[\\mathbb{Q}(e^{2\\pi i/n}):\\mathbb{Q}] \\leq 2$. Then we discuss how such convergence may fail in the case of $n=7$."},"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":"1502.02359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-02-09T04:57:32Z","cross_cats_sorted":[],"title_canon_sha256":"03e477d0dc072db9fc5510150e660ad7caf3334d4cac3a4d403c670651b3943a","abstract_canon_sha256":"48d4e36639f757541f2b8e5636970b979491987295b09733ca084b91b4c44663"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:42.729035Z","signature_b64":"2LY2o9kpyB9+RZutdzpKuXEW0WNsYzEqosct376y4DmvyYLMvmn7IJxz2JRi4Qgg0RN+I/oV7Bd9wiR0zG2hDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"546759bcf9accb22e653f3b28c3e647443535eaa0d7c15f5fecf560e2c2f5796","last_reissued_at":"2026-05-18T02:27:42.728265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:42.728265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Outer Billiards with Contraction: Regular Polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"In-Jee Jeong","submitted_at":"2015-02-09T04:57:32Z","abstract_excerpt":"We study outer billiards with contraction outside regular polygons. 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