{"paper":{"title":"When almost all sets are difference dominated in $\\mathbb{Z}/n\\mathbb{Z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Adam Lott, Anand Hemmady, Steven J. Miller","submitted_at":"2016-08-10T15:22:39Z","abstract_excerpt":"We investigate the behavior of the sum and difference sets of $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ chosen independently and randomly according to a binomial parameter $p(n) = o(1)$. We show that for rapidly decaying $p(n)$, $A$ is almost surely difference-dominated as $n \\to \\infty$, but for slowly decaying $p(n)$, $A$ is almost surely balanced as $n \\to \\infty$, with a continuous phase transition as $p(n)$ crosses a critical threshold. Specifically, we show that if $p(n) = o(n^{-1/2})$, then $|A-A|/|A+A|$ converges to $2$ almost surely as $n \\to \\infty$ and if $p(n) = c \\cdot n^{-1/2}$, then "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03209","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"}