The Kakeya set and maximal conjectures for algebraic varieties over finite fields

TL;DR

This paper uses the polynomial method to prove optimal estimates for Kakeya sets and maximal functions on algebraic varieties over finite fields, confirming the Kakeya maximal conjecture in this setting.

Contribution

It extends the polynomial method to algebraic varieties, establishing optimal Kakeya estimates and confirming the conjecture over finite fields.

Findings

Optimal Kakeya maximal estimates for algebraic varieties

Confirmation of the Kakeya maximal conjecture in finite fields

Strengthening of Dvir's results using algebraic geometry

Abstract

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective variety $W \subset \P^n(F)$, we establish the Kakeya maximal estimate $$ \| \sup_{\gamma \ni w} \sum_{v \in \gamma(F)} |f(v)| \|_{\ell^n(W)} \leq C_{n,W,d} |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$$ for all functions $f: F^n \to \R$ and $d \geq 1$, where for each $w \in W$, the supremum is over all irreducible algebraic curves in $F^n$ of degree at most $d$ that pass through $w$ but do not lie in $W$, and with $C_{n,W,d}$ depending only on $n, d$ and the degree of $W$; the special case when $W$ is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.