For a finite vector space V and a nonnegative integer r≥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⊆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 r-1, we have |V\K|=Θ(nq n-r+1); this improves the previously known bounds |V\K|=Ω(q n-r+1) and |V\K|=O(n 2 q n-r+1).
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
- Finite field
- Kakeya problem
- Kakeya set
- Polynomial method