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On directions determined by subsets of vector spaces over finite fields. (English) Zbl 1235.52027

The paper studies how many vectors guarantee in a \(d\)-dimensional vector space over a finite field with \(q\) elements that the differences of these vectors represent all possible directions in the space. More than \(q^{d-1}\) vectors suffice, and there is a characterization of sets of \(q^{d-1}\) vectors that fail this property. Similar results hold if one wants to guarantee the presence of all differences determined by a \(k\)-dimensional subspace. For subsets of the \(d\)-dimensional Euclidean plane, the analogous question was studied by J. Pach, R. Pinchasi and M. Sharir [Discrete Comput. Geom. 38, 399–441 (2007; Zbl 1146.52011)].

MSC:

52C10 Erdős problems and related topics of discrete geometry

Citations:

Zbl 1146.52011
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