{"paper":{"title":"Balancing Sets of Vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'abor Heged\\\"us","submitted_at":"2014-12-17T13:58:45Z","abstract_excerpt":"Let $n$ be an arbitrary integer, let $p$ be a prime factor of $n$. Denote by $\\omega_1$ the $p^{th}$ primitive unity root, $\\omega_1:=e^{\\frac{2\\pi i}{p}}$. Define $\\omega_i:=\\omega_1^i$ for $0\\leq i\\leq p-1$ and $B:=\\{1,\\omega_1,...,\\omega_{p-1}\\}^n$. Denote by $K(n,p)$ the minimum $k$ for which there exist vectors $v_1,...,v_k\\in B$ such that for any vector $w\\in B$, there is an $i$, $1\\leq i\\leq k$, such that $v_i\\cdot w=0$, where $v\\cdot w$ is the usual scalar product of $v$ and $w$. Gr\\\"obner basis methods and linear algebra proof gives the lower bound $K(n,p)\\geq n(p-1)$. Galvin posed th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5394","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"}