The small Kakeya sets in $T_2^{\ast }(\mathcal{C})$, $\mathcal{C}$ a conic

Abstract

A Kakeya set in the linear representation $T_2^{\ast }(\mathcal{C})$, $\mathcal{C}$ a non-singular conic, is the point set covered by a set of $q+1$ lines, one through each point of $\mathcal{C}$. In this article we classify the small Kakeya sets in $T_2^{\ast }(\mathcal{C})$. The smallest Kakeya sets have size $\lfloor \frac{3q^2+2q}{4}\rfloor$, and all Kakeya sets with weight less than $\lfloor \frac{3(q^2-1)}{4}\rfloor +q$ are classified: there are approximately $\sqrt{\frac{q}{2}}$ types.

References in corpus (2)

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