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Suppose $h_1(A,B)\\geq s$. Then we show that:\n  (a) if $||A|-|B||\\leq s$ and $|A|+|B|\\geq 4s^2-6s+3$, then $$|A+B|\\geq (2-\\frac 1 s)(|A|+|B|)-2s+1;$$\n  (b) if $|A|\\geq |B|+s$ and $|B|\\geq 2s^2-{7/2}s+{3/2}$, then $$|A+B|\\geq |A|+(3-\\frac 2 s)|B|-s;$$\n  (c) if $|"},"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":"0710.3127","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-10-16T17:01:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"144033182db034bc008f8887b579ff034b45c8a97687547692ad09fc141ab245","abstract_canon_sha256":"28ef7846a65ccca407518cf21f1b2c3a17eb9389dc58cd9721002fd3bd42a57d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:04:58.857982Z","signature_b64":"SfkkLQb2HiDifzAliaIwZM+2EkDZxX5qIiJYhm6xkyxaCm06URkdtiN4Uj+d+DlRh2b+iZ/daApgpMpKkTbNDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b606bdb33d97d10ef8f8d845538aaee7fa92bddcceb61b24dcee7c1c19a05311","last_reissued_at":"2026-07-04T15:04:58.857634Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:04:58.857634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Properties of two dimensional sets with small sumset","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"David J. Grynkiewicz, Oriol Serra","submitted_at":"2007-10-16T17:01:35Z","abstract_excerpt":"Let $A, B\\subseteq \\mathbb{R}^2$ be finite, nonempty subsets, let $s\\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines, $\\ell_1,...,\\ell_{t},\\ell'_1,...,\\ell'_{t}$, with $A\\subseteq \\bigcup_{i=1}^{t}\\ell_i$ and $B\\subseteq\\bigcup_{i=1}^{t}\\ell'_i$. Suppose $h_1(A,B)\\geq s$. Then we show that:\n  (a) if $||A|-|B||\\leq s$ and $|A|+|B|\\geq 4s^2-6s+3$, then $$|A+B|\\geq (2-\\frac 1 s)(|A|+|B|)-2s+1;$$\n  (b) if $|A|\\geq |B|+s$ and $|B|\\geq 2s^2-{7/2}s+{3/2}$, then $$|A+B|\\geq |A|+(3-\\frac 2 s)|B|-s;$$\n  (c) if $|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3127","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0710.3127/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"0710.3127","created_at":"2026-07-04T15:04:58.857695+00:00"},{"alias_kind":"arxiv_version","alias_value":"0710.3127v1","created_at":"2026-07-04T15:04:58.857695+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3127","created_at":"2026-07-04T15:04:58.857695+00:00"},{"alias_kind":"pith_short_12","alias_value":"WYDL3MZ5S7IQ","created_at":"2026-07-04T15:04:58.857695+00:00"},{"alias_kind":"pith_short_16","alias_value":"WYDL3MZ5S7IQ56HY","created_at":"2026-07-04T15:04:58.857695+00:00"},{"alias_kind":"pith_short_8","alias_value":"WYDL3MZ5","created_at":"2026-07-04T15:04:58.857695+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/WYDL3MZ5S7IQ56HY3BCVHCVO47","json":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47.json","graph_json":"https://pith.science/api/pith-number/WYDL3MZ5S7IQ56HY3BCVHCVO47/graph.json","events_json":"https://pith.science/api/pith-number/WYDL3MZ5S7IQ56HY3BCVHCVO47/events.json","paper":"https://pith.science/paper/WYDL3MZ5"},"agent_actions":{"view_html":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47","download_json":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47.json","view_paper":"https://pith.science/paper/WYDL3MZ5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0710.3127&json=true","fetch_graph":"https://pith.science/api/pith-number/WYDL3MZ5S7IQ56HY3BCVHCVO47/graph.json","fetch_events":"https://pith.science/api/pith-number/WYDL3MZ5S7IQ56HY3BCVHCVO47/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47/action/storage_attestation","attest_author":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47/action/author_attestation","sign_citation":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47/action/citation_signature","submit_replication":"https://pith.science/pith/WYDL3MZ5S7IQ56HY3BCVHCVO47/action/replication_record"}},"created_at":"2026-07-04T15:04:58.857695+00:00","updated_at":"2026-07-04T15:04:58.857695+00:00"}