{"paper":{"title":"Nested efficient congruencing and relatives of Vinogradov's mean value theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Trevor D. Wooley","submitted_at":"2017-08-03T17:10:29Z","abstract_excerpt":"We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\\varphi_j\\in \\mathbb Z[t]$ $(1\\le j\\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\\le k(k+1)/2$, then for all complex sequences $(\\mathfrak a_n)$, and for each $\\epsilon>0$, one has \\[ \\int_{[0,1)^k} \\left| \\sum_{|n|\\le X} {\\mathfrak a}_n e(\\alpha_1\\varphi_1(n)+\\ldots +\\alpha_k\\varphi_k(n)) \\right|^{2s} {\\rm d}{\\boldsymbol \\alpha} \\ll X^\\epsilon \\left( \\sum_{|n|\\le X} |{\\mathfrak a}_n|^2\\right)^s. \\] As a special case of this result, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01220","kind":"arxiv","version":2},"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"}