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Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatori"},"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":"1806.04267","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-11T23:28:32Z","cross_cats_sorted":["math.CO","math.DS"],"title_canon_sha256":"ebd013ed4e6fcf014d013dc137fcdd88ca489209fc0a33ff074b64088edb536b","abstract_canon_sha256":"d63266035d6cbd7ba70dccdec3a39409da26448cbd3c924babf362bd9c873d46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:13.421293Z","signature_b64":"YYQlLqJTZ/+3RwudZQZiNfShULAuXCRI/0cxLyC4onVTPvGPuMBQKX0Jz0rDGUsHFyE6phkH6r+cnOA5vdOCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a75c28cf35fc6406a62113a1a4051570bf6b72d726fb671bf472dce414ba5a8c","last_reissued_at":"2026-05-17T23:50:13.420626Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:13.420626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On uniformity of $q$-multiplicative sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.DS"],"primary_cat":"math.NT","authors_text":"Aihua Fan, Jakub Konieczny","submitted_at":"2018-06-11T23:28:32Z","abstract_excerpt":"We show that any $q$-multiplicative sequence which is \\emph{oscillating} of order $1$, i.e.\\ does not correlate with linear phase functions $e^{2\\pi i n\\alpha}$ ($\\alpha \\in \\mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\\pi i p(n)}$ ($p \\in \\mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. 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