{"paper":{"title":"Some properties of lower level-sets of convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ernie Croot","submitted_at":"2011-08-07T21:55:46Z","abstract_excerpt":"In the present paper we prove a certain lemma about the structure of \"lower level-sets of convolutions\", which are sets of the form $\\{x \\in \\Z_N : 1_A*1_A(x) \\leq \\gamma N\\}$ or of the form $\\{x \\in \\Z_N : 1_A*1_A(x) < \\gamma N\\}$, where $A$ is a subset of $\\Z_N$. One result we prove using this lemma is that if $|A| = \\theta N$ and $|A+A| \\leq (1-\\eps) N$, $0 < \\eps < 1$, then this level-set contains an arithmetic progression of length at least $N^c$, $c = c(\\theta, \\eps,\\gamma) > 0$. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.1578","kind":"arxiv","version":4},"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"}