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On dyadic intervals $X\\le n<2X$, put $L=\\log(2X)$ and ${\\mathcal F}_X=\\sqrt2(\\log2)^{1/4}L^{1/4}\\exp\\sqrt{(\\log2)L}$. Uniformly for $1\\le z\\le Z(X)=o(L^{1/4})$, we prove ${\\mathbb P}_X(A(n)\\le {\\mathcal F}_X\\exp(-z))\\asymp \\exp(-2z)$ and ${\\mathbb P}_X(A(n)>{\\mathcal F}_X\\exp(z))\\ll \\exp(-2z)$. Consequently $\\log A(n)=\\sqrt{(\\log2)\\log n}+\\frac14\\log\\log n+O_{\\rm dens}(1)$. We also prove dyadic nonconcentration: no scalar center","authors_text":"Eric Li (Trinity College, University of Cambridge)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-27T19:39:59Z","title":"A Resolution of Erd\\H{o}s Problem 731 under Dyadic Regularity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29062","kind":"arxiv","version":1},"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":"sha256:379052fb8dd142094d70b89bbabc5f251fc3cd0328306fe5f5713e59594d4de9","target":"record","created_at":"2026-06-30T01:17:51Z","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":"a53a3f692a6de9572d3edf570f97471b2979cc0fe33e3d227578f101fd63b5b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-27T19:39:59Z","title_canon_sha256":"5fe111cc6f5c18f59faef1eb77dd6babb070e5d3aef55a45bdae8de45884f548"},"schema_version":"1.0","source":{"id":"2606.29062","kind":"arxiv","version":1}},"canonical_sha256":"c931b05cf14249e46b816dacc27c505c939f2c588603c8b8be55710918054b6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c931b05cf14249e46b816dacc27c505c939f2c588603c8b8be55710918054b6c","first_computed_at":"2026-06-30T01:17:51.253533Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T01:17:51.253533Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q4XahJ6UdT+Z33lBQzFrHhjgpbNTlj8/iIH1utvJPJhIXTSTx9EKhdDTb/CAw92HkZYmJRAIBSkolV4GIq4+CA==","signature_status":"signed_v1","signed_at":"2026-06-30T01:17:51.254020Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.29062","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:379052fb8dd142094d70b89bbabc5f251fc3cd0328306fe5f5713e59594d4de9","sha256:0c0fa9b041399916473d83da141b4615d79175ca1b074392cb6eb7a0a35f2b24"],"state_sha256":"b904baa4615f50bd1e2baf9d26e080f1dab8929884f5c5187b4e51144f22f5f6"}