{"paper":{"title":"An Integral Mean Value Theorem for Weyl Sums over Broken Arcs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"YaoJie Guo","submitted_at":"2026-08-05T15:15:22Z","abstract_excerpt":"In this article, we research the mean value of integral of exponential sum $S(\\alpha)=\\sum_{u\\in I}e(\\alpha u^k)$, where $I$ is a short interval whose length is $2N^{\\theta},\\theta<1$, and the broken arc $\\mathfrak{m^*}$ is a subset of a following minor arcs \\[ \\mathfrak{m}=\\bigcap_{j\\leq k-1}\\left\\{\\alpha:\\forall q<(\\log N)^A,h<q,(h,q)=1,\\Big|\\alpha-\\frac{h}{q}\\Big|>\\frac{1}{qN^{(k-j-1/2)\\theta}}\\right\\} \\] which has measure at least $c>0$. By setting $m$ is a sufficiently large number, $N$ is be sufficiently large in terms of $m$. When $k>\\log m$ and $\\frac{\\log k}{\\log m}<1/2$ we set the fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04948","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.04948/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}