# Conical Kakeya and Nikodym Sets in Finite Fields

## TL;DR

This paper extends Dvir's polynomial method to establish size lower bounds for Kakeya and Nikodym sets in finite fields involving conics like parabolae and hyperbolae, providing explicit bounds and constants.

## Contribution

It generalizes the finite field Kakeya problem to conic sets, including explicit bounds and constants, and introduces bounds for conical Nikodym sets.

## Key findings

- Lower bounds for conical Kakeya sets with explicit constants
- Lower bounds for conical Nikodym sets including ellipses
- Extension of polynomial method to non-degenerate conics 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.