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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.5912","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-05-26T19:22:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2bb72d23e1628a9bc589408755ef0c6ed2fc63ec23861217d12b12cb04ce3360","abstract_canon_sha256":"aea6d53b90f1779ebd473f74998ed86d62d3cff8b63e696215cfdd7c4597b40a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:48.758187Z","signature_b64":"S1Bc/iTzfA2lQyYFXh5E0xgnOAGWB32KH0+UMGwudannodQfEvD2mEr8zhB3QUgt6bGspMwhGtnX7EO7706rBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9774f4755495f546cb80beb1fc8b0d2f584df110c871f4fd178ca68c2acd955","last_reissued_at":"2026-05-18T03:54:48.757553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:48.757553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.5912","created_at":"2026-05-18T03:54:48.757659+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5912v1","created_at":"2026-05-18T03:54:48.757659+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5912","created_at":"2026-05-18T03:54:48.757659+00:00"},{"alias_kind":"pith_short_12","alias_value":"3F3U6R2VJFPV","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"3F3U6R2VJFPVI3FY","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"3F3U6R2V","created_at":"2026-05-18T12:26:50.516681+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L","json":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L.json","graph_json":"https://pith.science/api/pith-number/3F3U6R2VJFPVI3FYBPVR7SFQ2L/graph.json","events_json":"https://pith.science/api/pith-number/3F3U6R2VJFPVI3FYBPVR7SFQ2L/events.json","paper":"https://pith.science/paper/3F3U6R2V"},"agent_actions":{"view_html":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L","download_json":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L.json","view_paper":"https://pith.science/paper/3F3U6R2V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5912&json=true","fetch_graph":"https://pith.science/api/pith-number/3F3U6R2VJFPVI3FYBPVR7SFQ2L/graph.json","fetch_events":"https://pith.science/api/pith-number/3F3U6R2VJFPVI3FYBPVR7SFQ2L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L/action/storage_attestation","attest_author":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L/action/author_attestation","sign_citation":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L/action/citation_signature","submit_replication":"https://pith.science/pith/3F3U6R2VJFPVI3FYBPVR7SFQ2L/action/replication_record"}},"created_at":"2026-05-18T03:54:48.757659+00:00","updated_at":"2026-05-18T03:54:48.757659+00:00"}