{"paper":{"title":"The PFR Conjecture Holds for Two Opposing Special Cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Thomas Holenstein","submitted_at":"2013-11-01T13:06:44Z","abstract_excerpt":"Let $A \\subseteq F_2^n$ be a set with $|2A| = K|A|$. We prove that if (1) for at least a fraction $1-K^{-9}$ of all $s \\in 2A$, the set $(A+s) \\cap A$ has size at most $L\\cdot|A|/K$, or (2) for at least a fraction $K^{-L}$ of all $s \\in 2A$, the set $(A+s) \\cap A$ has size at least $|A|\\cdot(1- K^{-1/L})$, then there is a subset $B \\subseteq A$ of size $|A|/K^{O_L(1)}$ such that $\\mathrm{span}(B) \\leq K^{O_L(1)}\\cdot|A|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0172","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"}