Geodesic
Kakeya-type sets in finite vector spaces
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 3, pp. 337-355.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: For a finite vector space $V$ and a nonnegative integer $r\leq $dim $V$, we estimate the smallest possible size of a subset of $V$, containing a translate of every $r$-dimensional subspace. In particular, we show that if $K\subseteq V$ is the smallest subset with this property, $n$ denotes the dimension of $V$, and $q$ is the size of the underlying field, then for $r$ bounded and $r n\leq rq ^{ r - 1}$, we have | $V\setminus K|= \Theta ( nq ^{ n - r+1})$; this improves the previously known bounds | $V\setminus K|= \Omega ( q ^{ n - r+1})$ and | $V\setminus K|= O( n ^{2} q ^{ n - r+1})$.
Keywords: keywords Kakeya set, Kakeya problem, polynomial method, finite field 
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     author = {Kopparty, Swastik and Lev, Vsevolod F. and Saraf, Shubhangi and Sudan, Madhu},
     title = {Kakeya-type sets in finite vector spaces},
     journal = {Journal of Algebraic Combinatorics},
     pages = {337--355},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2011__34_3_a7/}
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Kopparty, Swastik; Lev, Vsevolod F.; Saraf, Shubhangi; Sudan, Madhu. Kakeya-type sets in finite vector spaces. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 3, pp. 337-355. http://geodesic.mathdoc.fr/item/JAC_2011__34_3_a7/