{"paper":{"title":"Bounds for generalized Sidon sets","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Craig Timmons, Rafael Tesoro, Xing Peng","submitted_at":"2013-11-12T23:59:48Z","abstract_excerpt":"Let $\\Gamma$ be an abelian group and $g \\geq h \\geq 2$ be integers. A set $A \\subset \\Gamma$ is a $C_h[g]$-set if given any set $X \\subset \\Gamma$ with $|X| = k$, and any set $\\{ k_1 , \\dots , k_g \\} \\subset \\Gamma$, at least one of the translates $X+ k_i$ is not contained in $A$. For any $g \\geq h \\geq 2$, we prove that if $A \\subset \\{1,2, \\dots ,n \\}$ is a $C_h[g]$-set in $\\mathbb{Z}$, then $|A| \\leq (g-1)^{1/h} n^{1 - 1/h} + O(n^{1/2 - 1/2h})$. We show that for any integer $n \\geq 1$, there is a $C_3 [3]$-set $A \\subset \\{1,2, \\dots , n \\}$ with $|A| \\geq (4^{-2/3} + o(1)) n^{2/3}$. We als"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2985","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"}