منابع مشابه
Kakeya-type sets in finite vector spaces
For a finite vector space V and a non-negative integer r ≤ dim V we estimate the smallest possible size of a subset of V , containing a translate of every rdimensional subspace. In particular, we show that if K ⊆ 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 ≤ rq we have |V \K| = Θ(nq); this im...
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We compute the Minkowski dimension for a family of self-affine sets on R. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.
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For a finite vector space V and a non-negative integer r ≤ dim V we estimate the smallest possible size of a subset of V , containing a translate of every rdimensional subspace. In particular, we show that if K ⊆ 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 ≤ rq we have |V \K| = Θ(nq); this im...
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We give improved lower bounds on the size of Kakeya and Nikodym sets over Fq. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in Fq, and show that this conjecture implies an optimal bound on the size of a Nikodym set. Finally, we study the notion of a weak Nikodym set and give improved, and in some special cases optimal, bounds fo...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2011
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2010.03.007