Finite Field Kakeya and Nikodym Sets in Three Dimensions

We give improved lower bounds on the size of Kakeya and Nikodym sets over $\mathbb{F}_q^3$. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in $\mathbb{F}_q^3$, 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 for weak Nikodym sets in $\mathbb{F}_q^2$ and $\mathbb{F}_q^3$.