{"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"}