{"paper":{"title":"On sumsets in ${\\Bbb F}_2^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Chaohua Jia","submitted_at":"2012-05-26T19:22:33Z","abstract_excerpt":"Let ${\\Bbb F}_2$ be the finite field of two elements, ${\\Bbb F}_2^n$ be the vector space of dimension $n$ over ${\\Bbb F}_2$. For sets $A,\\,B\\subseteq{\\Bbb F}_2^n$, their sumset is defined as the set of all pairwise sums $a+b$ with $a\\in A,\\,b\\in B$.\n  Ben Green and Terence Tao proved that, let $K\\geq 1$, if$A,\\,B\\subseteq{\\Bbb F}_2^n$ and $|A+B|\\leq K|A|^{1\\over 2}|B|^{1\\over 2}$, then there exists a subspace $H\\subseteq{\\Bbb F}_2^n$ with $$ |H|\\gg\\exp(-O(\\sqrt{K}\\log K))|A| $$ and $x,\\,y\\in{\\Bbb F}_2^n$ such that $$ |A\\cap(x+H)|^{1\\over 2}|B\\cap(y+H)|^{1\\over 2}\\geq{1\\over 2K}|H|. $$\n  In thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5912","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"}