{"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"}