{"paper":{"title":"$\\boldsymbol L^{\\boldsymbol 1}$-Norm of Steinhaus chaose on the polydisc","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Michel Weber","submitted_at":"2014-11-26T16:33:01Z","abstract_excerpt":"Let $J_n\\subset[1,n]$, $n=1,2,\\ldots$ be increasing sets of mutually coprime numbers. Under reasonable conditions on the coefficient sequence $\\{c^j_n\\}_{n,j}$, we show that $$ \\lim_{T\\to \\infty}\\frac{1}{T} \\int_{0}^T \\Big| \\sum_{j\\in J_n} c^j_n\\,j^{it}\\Big| \\dd t\\sim \\big(\\frac{\\pi} {2}\\sum_{j\\in J_n} (c^j_n)^2\\big)^{1/2} $$ as $n\\to \\infty$. We also show by means of an elementary device that for all $0<\\a<2$, \\begin{eqnarray*}\n  \\lim_{T\\to \\infty} \\Big(\\frac{1}{T} \\int_{0}^T \\big| \\sum_{n=1}^N n^{-it}\\big|^\\a\\dd t\\Big)^{1/\\a} \\ge C_\\a\\, \\frac{ N^{\\frac{1}{2}}} {\\big( \\log N\\big)^{{\\frac{1}{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.7291","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"}