A note on large Kakeya sets

Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q2 – q q $\begin{array}{} \displaystyle \sqrt{q} \end{array}$ + 32 $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.