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

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

2009-03-10

arxiv.org/abs/0903.1879v2

Mathematics

Combinatorics

25 pages, no figures, to appear, Mathematika. This is the final version, incorporating the referee report

Scientific paper

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.

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