{"paper":{"title":"Order-preserving Freiman isomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Albert Bush, Ernie Croot, Gagik Amirkhanyan","submitted_at":"2014-09-30T13:16:44Z","abstract_excerpt":"An order-preserving Freiman 2-isomorphism is a map $\\phi:X \\rightarrow \\mathbb{R}$ such that $\\phi(a) < \\phi(b)$ if and only if $a < b$ and $\\phi(a)+\\phi(b) = \\phi(c)+\\phi(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \\in X$. We show that for any $A \\subseteq \\mathbb{Z}$, if $|A+A| \\le K|A|$, then there exists a subset $A' \\subseteq A$ such that the following holds: $|A'| \\gg_K |A|$ and there exists an order-preserving Freiman 2-isomorphism $\\phi: A' \\rightarrow [-c|A|,c|A|] \\cap \\mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8535","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"}