{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:BXDEMCOLPM74BWOC56LCT37DWC","short_pith_number":"pith:BXDEMCOL","schema_version":"1.0","canonical_sha256":"0dc64609cb7b3fc0d9c2ef9629efe3b0bdc7474d3b2bc73b2216bccef9a117fd","source":{"kind":"arxiv","id":"1611.01739","version":4},"attestation_state":"computed","paper":{"title":"Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2016-11-06T07:53:24Z","abstract_excerpt":"We consider the space $A(\\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\\mathbb{T}^d$. The norm on $A(\\mathbb{T}^d)$ is naturally defined by $\\|f\\|_{A}=\\|\\widehat{f}\\|_{l^1}$, where $\\widehat{f}$ is the Fourier transform of a function $f$. For real functions $\\varphi$ of a certain special form on $\\mathbb T^d, \\,d\\geq 2,$ we obtain lower bounds for the norms $\\|e^{i\\lambda\\varphi}\\|_A$ as $\\lambda\\rightarrow\\infty$. In particular, we show that if $\\varphi(x, y)=a(x)|y|$ for $|y|\\leq\\pi$, where $a\\in A(\\mathbb{T})$ is an arbitrary nonconstant real function, then $\\|e^{i\\la"},"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":"1611.01739","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-11-06T07:53:24Z","cross_cats_sorted":[],"title_canon_sha256":"0a39337ffcdeaa9c518c4afba25fafe5608bc5871b961039671418106cd7e36a","abstract_canon_sha256":"a3c29df6632c234518328c3e766bc971057458c4b40c1f0176bbb8346c78b634"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:53.246998Z","signature_b64":"LDhL51Z3bf+7cWUuTnorWPVR2+UgWMUv/3908zQAN/E3MdXqvX+NV/5XWpXcKS7ds7UKTDO56wHTv/K+uNlTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0dc64609cb7b3fc0d9c2ef9629efe3b0bdc7474d3b2bc73b2216bccef9a117fd","last_reissued_at":"2026-05-17T23:48:53.246477Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:53.246477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Vladimir Lebedev","submitted_at":"2016-11-06T07:53:24Z","abstract_excerpt":"We consider the space $A(\\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\\mathbb{T}^d$. The norm on $A(\\mathbb{T}^d)$ is naturally defined by $\\|f\\|_{A}=\\|\\widehat{f}\\|_{l^1}$, where $\\widehat{f}$ is the Fourier transform of a function $f$. For real functions $\\varphi$ of a certain special form on $\\mathbb T^d, \\,d\\geq 2,$ we obtain lower bounds for the norms $\\|e^{i\\lambda\\varphi}\\|_A$ as $\\lambda\\rightarrow\\infty$. In particular, we show that if $\\varphi(x, y)=a(x)|y|$ for $|y|\\leq\\pi$, where $a\\in A(\\mathbb{T})$ is an arbitrary nonconstant real function, then $\\|e^{i\\la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01739","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1611.01739","created_at":"2026-05-17T23:48:53.246563+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.01739v4","created_at":"2026-05-17T23:48:53.246563+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.01739","created_at":"2026-05-17T23:48:53.246563+00:00"},{"alias_kind":"pith_short_12","alias_value":"BXDEMCOLPM74","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BXDEMCOLPM74BWOC","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BXDEMCOL","created_at":"2026-05-18T12:30:09.641336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC","json":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC.json","graph_json":"https://pith.science/api/pith-number/BXDEMCOLPM74BWOC56LCT37DWC/graph.json","events_json":"https://pith.science/api/pith-number/BXDEMCOLPM74BWOC56LCT37DWC/events.json","paper":"https://pith.science/paper/BXDEMCOL"},"agent_actions":{"view_html":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC","download_json":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC.json","view_paper":"https://pith.science/paper/BXDEMCOL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.01739&json=true","fetch_graph":"https://pith.science/api/pith-number/BXDEMCOLPM74BWOC56LCT37DWC/graph.json","fetch_events":"https://pith.science/api/pith-number/BXDEMCOLPM74BWOC56LCT37DWC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC/action/storage_attestation","attest_author":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC/action/author_attestation","sign_citation":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC/action/citation_signature","submit_replication":"https://pith.science/pith/BXDEMCOLPM74BWOC56LCT37DWC/action/replication_record"}},"created_at":"2026-05-17T23:48:53.246563+00:00","updated_at":"2026-05-17T23:48:53.246563+00:00"}