{"paper":{"title":"Difference Sets and Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Alex Rice, Neil Lyall","submitted_at":"2015-04-20T00:50:05Z","abstract_excerpt":"We provide upper bounds on the largest subsets of $\\{1,2,\\dots,N\\}$ with no differences of the form $h_1(n_1)+\\cdots+h_{\\ell}(n_{\\ell})$ with $n_i\\in \\mathbb{N}$ or $h_1(p_1)+\\cdots+h_{\\ell}(p_{\\ell})$ with $p_i$ prime, where $h_i\\in \\mathbb{Z}[x]$ lie in in the classes of so-called intersective and $\\mathcal{P}$-intersective polynomials, respectively. For example, we show that a subset of $\\{1,2,\\dots,N\\}$ free of nonzero differences of the form $n^j+m^k$ for fixed $j,k\\in \\mathbb{N}$ has density at most $e^{-(\\log N)^{\\mu}}$ for some $\\mu=\\mu(j,k)>0$. Our results, obtained by adapting two Fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04904","kind":"arxiv","version":3},"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"}