{"paper":{"title":"An extension of the direction problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marcella Tak\\'ats, P\\'eter Sziklai","submitted_at":"2014-07-21T19:55:45Z","abstract_excerpt":"Let $U$ be a point set in the $n$-dimensional affine space ${\\rm AG}(n,q)$ over the finite field of $q$ elements and $0\\leq k\\leq n-2$. In this paper we extend the definition of directions determined by $U$: a $k$-dimensional subspace $S_k$ at infinity is determined by $U$ if there is an affine $(k+1)$-dimensional subspace $T_{k+1}$ through $S_k$ such that $U\\cap T_{k+1}$ spans $T_{k+1}$. We examine the extremal case $|U|=q^{n-1}$, and classify point sets NOT determining every $k$-subspace in certain cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5613","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"}