{"paper":{"title":"On product of difference sets for sets of positive density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Alexander Fish","submitted_at":"2017-02-08T17:55:14Z","abstract_excerpt":"In this paper we prove that given two sets $E_1,E_2 \\subset \\mathbb{Z}$ of positive density, there exists $k \\geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\\mathbb{Z} \\subset (E_1-E_1)\\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $\\alpha,\\beta > 0$ then there exist $N_0$ and $d_0$ which depend only on $\\alpha$ and $\\beta$ such that for every $N \\geq N_0$ and $E_1,E_2 \\subset \\mathbb{Z}_N$ with $|E_1| \\geq \\alpha N, |E_2| \\geq \\beta N$ there exists $d \\leq d_0$ a divisor of $N$ satisfying $d \\, \\mathbb{Z}_N \\subset (E_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02544","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"}