Improved lower bound on the size of Kakeya sets over finite fields

In a recent breakthrough, Dvir showed that every Kakeya set in $\F^n$ must be of cardinality at least $c_n |\F|^n$ where $c_n \approx 1/n!$. We improve this lower bound to $\beta^n |\F|^n$ for a constant $\beta > 0$. This pins down the growth of the leading constant to the right form as a function of $n$.