The small Kakeya sets in $T^{*}_{2}(\mathcal{C})$, $\mathcal{C}$ a conic

TL;DR

This paper classifies small Kakeya sets in a geometric structure related to a conic, identifying the smallest such sets and categorizing those below a certain size threshold.

Contribution

It provides a complete classification of small Kakeya sets in $T^{*}_{2}( ext{C})$, including the smallest sets and a near-complete categorization of sets below a specific size.

Findings

Smallest Kakeya sets have size approximately (3q^2+2q)/4.

All Kakeya sets with weight less than a certain threshold are classified.

Approximately sqrt(q/2) types of such sets are identified.

Abstract

A Kakeya set in the linear representation $T^{*}_{2}(\mathcal{C})$, $\mathcal{C}$ 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 $T^{*}_{2}(\mathcal{C})$. The smallest Kakeya sets have size $\left\lfloor\frac{3q^{2}+2q}{4}\right\rfloor$, and all Kakeya sets with weight less than $\left\lfloor\frac{3(q^{2}-1)}{4}\right\rfloor+q$ are classified: there are approximately $\sqrt{\frac{q}{2}}$ types.