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We prove that for $l \\to \\infty$,\n  $$|\\mathcal{N}_l| = \\mathcal{O}\\left(\\left\\lfloor \\tfrac{l+1}{2} \\right\\rfloor^{s-1} \\rho^{\\left\\lfloor \\tfrac{l+1}{2} \\right\\rfloor} \\right),\n  $$\n  where $\\mathcal O$ is the `big O', $\\rho \\in [\\sqrt{2}, 2]$ is the unique positive real root of\n  $$\n  p(x) = x^{r+1} - 2\\sum_{j=1}^{r-1} x^{r-j} - 1,\n  $$\n  and $s$ is the maximal multiplicity among the roots of $p(x)$.\n  Our method relies o","authors_text":"Debattam Das, Krishnendu Gongopadhyay","cross_cats":["math.GT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-11-01T17:05:22Z","title":"Spherical growth of reciprocal classes in the Hecke Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.00739","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e71a296a3c15cc81f899aa4682d417aeda391b1926e078f43585879c2bced1f3","target":"record","created_at":"2026-07-05T11:09:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"223b8e6dd15a06df98ef243e80de01c4161a70a0b99e7f4fd546f0239e0f2d04","cross_cats_sorted":["math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-11-01T17:05:22Z","title_canon_sha256":"7e3a310e013d3d520474c4c4faa87c25870d068d6bcf2850abfe596e3586423e"},"schema_version":"1.0","source":{"id":"2411.00739","kind":"arxiv","version":2}},"canonical_sha256":"4739b6032700aac00ad7ffb1a2c56e0f96f47ffc4be9054206bf8fb0120e72d6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4739b6032700aac00ad7ffb1a2c56e0f96f47ffc4be9054206bf8fb0120e72d6","first_computed_at":"2026-07-05T11:09:43.770309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:09:43.770309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iE/yLyDawCOdjg4tY2zrxl8YmligRsHVOn2AemgCYZRhtOQyVNFsafLY6Ii7g6mEN5m2KmP2wureNr3iv59OAA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:09:43.770839Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.00739","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e71a296a3c15cc81f899aa4682d417aeda391b1926e078f43585879c2bced1f3","sha256:0884ca0e5c23d3e0978ec3883d35552cd90f5182988393ce93f02ba554e1b9be"],"state_sha256":"d677e8538a7f8dba0a4a9c6af53ddd9b7f3ead6bc979a5e9e4cce32886768ac5"}