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We prove that, for Lebesgue-almost every \\(x\\), \\[\n  \\frac{1}{2}\n  \\le \\liminf_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}\n  \\le \\limsup_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}\n  \\le \\frac{q+1}{q-1}\\,. \\] If, in addition, \\(a_n\\mid a_{n+1}\\) for every \\(n\\), then this can be improved to \\[\n  \\lim_{N\\to\\infty}\\frac{NG_N(x)}{\\log N}=1 \\] for Lebesgue-almost every \\(x\\)."},"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":"2606.28860","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-27T10:54:00Z","cross_cats_sorted":["math.DS","math.PR"],"title_canon_sha256":"ff9920bf6a6537c6aba3d68cc2626ee18ee3bee16261320ccdaff4f75f6e79c9","abstract_canon_sha256":"2a043397b8680902f8a6e243f2772d995a20c258038ea4e642df18fa114aaca8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:16:54.904910Z","signature_b64":"Jw/CzgUEbCVzwoM5GY07V051DCHGtgtnmhfXk93XBDaB6H4J19jgMm9CwRAEAH3b3cUooJncjxiBZ+eJadp5Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5268d29ebc9019f4f917bdb04d563c60bad1945a3ed946d31d5294c6179cd58","last_reissued_at":"2026-06-30T01:16:54.904240Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:16:54.904240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximal Gaps for Dilated Lacunary Integer Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.PR"],"primary_cat":"math.NT","authors_text":"Bohan Yang, Yuval Peres","submitted_at":"2026-06-27T10:54:00Z","abstract_excerpt":"Let \\((a_n)_{n\\ge1}\\subset\\mathbb{N}\\) be a lacunary sequence, \\(a_{n+1}\\ge q a_n\\) for \\(q>1\\). 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