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The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make progress on fundamental questions concerning the iteration of $f$. In this paper we establish new relationships between these three concepts.\n  We prove that a certain weak regularity condition is necessary and sufficient for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\\neq A(f)$.\n  We also apply a result of Beurling that relates the size of the m"},"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":"1301.2193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-01-10T17:16:38Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"fa7a57bda30b093d4bab476555567c104d471ab8e5c348c1763fe2bfa9efdd05","abstract_canon_sha256":"6bafd4b2af259ce2bec14c6ba27c5bea6e9514e7b94595842a8bae2ea8958610"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:47.238207Z","signature_b64":"2xuv+npeIUP4/gz9qA9wAy+79CDu0SLhIXWAf5LHFuh5DCCHjyKjRgw1eTbm9sCE6x4MBX+NFAgdTDSPnhJ9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49aed34a07de1314fc36b7f7cb557c68a982d3268cbda4ab1ac48ef220be53fc","last_reissued_at":"2026-05-18T03:36:47.237550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:47.237550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity and fast escaping points of entire functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Gwyneth M. 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