{"paper":{"title":"Pointwise convergence of almost periodic Fourier series and associated series of dilates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christophe Cuny, Michel Weber","submitted_at":"2016-07-07T15:02:35Z","abstract_excerpt":"Let $\\mathcal S^2$ be the Stepanov space and let $ \\lambda_n\\uparrow\\infty$. Let $(a_n)_{n\\ge 1}$ be satisfying Wiener's condition $A:= \\sum_{n\\ge 1} \\big(\\sum_{k\\, :\\, n\\le \\lambda_k \\le n+1}|a_k|\\big)^2 <\\infty$. We prove that $\\big\\| \\sup_{N\\ge 1} \\big|\\sum_{n=1}^Na_n{\\rm e}^{i\\lambda_n t}\\big| \\, \\big\\|_{\\mathcal S^2}\\le C\\, A^{1/2} $ where $C>0$ denotes a universal constant. Moreover, the series $\\sum_{n\\ge 1} a_n{\\rm e}^{it\\lambda_n }$ converges for $\\lambda$-a.e. $t\\in \\mathbb R$. This contains as a special case Hedenmalm and Saksman result for Dirichlet series. We also obtain maximal i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02041","kind":"arxiv","version":1},"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"}