Conical Kakeya and Nikodym sets in finite fields

Abstract

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.)Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

The Spherical Kakeya Problem in Finite Fields

“…Several variants of Kakeya sets over finite fields have been studied as well, see for example [5]. In particular the paper [15] deals with conical Kakeya sets over finite fields, that is, subsets of F n q containing either a parabola or a hyperbola in every direction (ellipses are not used since they do not have a direction). By 'directions' we usually mean points of the hyper-plane at infinity lying on an object.…”