{"paper":{"title":"Cancellation in additively twisted sums on $\\mathrm{GL}(2)$ with non-linear phase","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yongxiao Lin, Zhi Qi","submitted_at":"2019-06-13T05:38:59Z","abstract_excerpt":"Let $\\lambda_g (n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for $\\mathrm{SL}_2 (\\mathbb{Z})$. The aim of this article is to get non-trivial bound on non-linearly additively twisted sums of the Fourier coefficients $\\lambda_g (n)$. Precisely, we prove for any $3/4 < \\beta < 3/2$, $\\beta \\neq 1 $, the following non-trivial estimate $$ \\sum_{n \\leq N}\\lambda_g(n)\\,e(\\alpha\\, n^{\\beta})\\ll_{g, \\alpha, \\beta, \\varepsilon} N^{\\frac{1}{2}+ \\frac{\\beta}{3} +\\varepsilon} + N^{\\frac{3}{2}-\\frac {2\\beta}{3} + \\varepsilon}, $$ for any $\\varepsilon > 0$. This is the first time th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.06371","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"}