An extension of the direction problem

Péter Sziklai; Marcella Takáts

doi:10.1016/j.disc.2012.02.021

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arXiv · 1407.5613

An extension of the direction problem

Péter Sziklai·Marcella Takáts

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Abstract

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.

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BibTeX RIS

Péter Sziklai, Marcella Takáts. 2014-07-21. An extension of the direction problem. https://doi.org/10.1016/j.disc.2012.02.021

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