{"paper":{"title":"Polynomial values modulo primes on average and sharpness of the larger sieve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xuancheng Shao","submitted_at":"2014-09-25T05:33:39Z","abstract_excerpt":"This paper is motivated by the following question in sieve theory. Given a subset $X\\subset [N]$ and $\\alpha\\in (0,1/2)$. Suppose that $|X\\pmod p|\\leq (\\alpha+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\\ll_{\\alpha}N^{\\alpha}$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\\ll_{\\alpha}N^{O(\\alpha^{2014})}$ for small $\\alpha$). The result follows from studying the average size of $|X\\pmod p|$ as $p$ varies, when $X=f(\\mathbb{Z})\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7160","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"}