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Jordan Ellenberg, Richard Oberlin, and I have just uploaded to the arXiv the paper “The Kakeya set and maximal conjectures for algebraic varieties over finite fields“, submitted to Mathematika.  This paper builds upon some work of Dvir and later authors on the Kakeya problem in finite fields, which I have discussed in this earlier blog post.  Dvir established the following:

Kakeya set conjecture for finite fields. Let F be a finite field, and let E be a subset of $F^n$ that contains a line in every direction.  Then E has cardinality at least $c_n |F|^n$ for some $c_n > 0$.

The initial argument of Dvir gave $c_n = 1/n!$.  This was improved to $c_n = c^n$ for some explicit $0 < c < 1$ by Saraf and Sudan, and recently to $c_n =1/2^n$ by Dvir, Kopparty, Saraf, and Sudan, which is within a factor 2 of the optimal result.

In our work we investigate a somewhat different set of improvements to Dvir’s result.  The first concerns the Kakeya maximal function $f^*: {\Bbb P}^{n-1}(F) \to {\Bbb R}$ of a function $f: F^n \to {\Bbb R}$, defined for all directions $\xi \in {\Bbb P}^{n-1}(F)$ in the projective hyperplane at infinity by the formula

$f^*(\xi) = \sup_{\ell // \xi} \sum_{x \in \ell} |f(x)|$

where the supremum ranges over all lines $\ell$ in $F^n$ oriented in the direction $\xi$.  Our first result is the endpoint $L^p$ estimate for this operator, namely

Kakeya maximal function conjecture in finite fields. We have $\| f^* \|_{\ell^n({\Bbb P}^{n-1}(F))} \leq C_n |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$ for some constant $C_n > 0$.

This result implies Dvir’s result, since if f is the indicator function of the set E in Dvir’s result, then $f^*(\xi) = |F|$ for every $\xi \in {\Bbb P}^{n-1}(F)$.  However, it also gives information on more general sets E which do not necessarily contain a line in every direction, but instead contain a certain fraction of a line in a subset of directions.  The exponents here are best possible in the sense that all other $\ell^p \to \ell^q$ mapping properties of the operator can be deduced (with bounds that are optimal up to constants) by interpolating the above estimate with more trivial estimates.  This result is the finite field analogue of a long-standing (and still open) conjecture for the Kakeya maximal function in Euclidean spaces; we rely on the polynomial method of Dvir, which thus far has not extended to the Euclidean setting (but note the very interesting variant of this method by Guth that has established the endpoint multilinear Kakeya maximal function estimate in this setting, see this blog post for further discussion).

It turns out that a direct application of the polynomial method is not sufficient to recover the full strength of the maximal function estimate; but by combining the polynomial method with the Nikishin-Maurey-Pisier-Stein “method of random rotations” (as interpreted nowadays by Stein and later by Bourgain, and originally inspired by the factorisation theorems of Nikishin, Maurey, and Pisier), one can already recover a “restricted weak type” version of the above estimate.  If one then enhances the polynomial method with the “method of multiplicities” (as introduced by Saraf and Sudan) we can then recover the full “strong type” estimate; a few more details below the fold.

It turns out that one can generalise the above results to more general affine or projective algebraic varieties over finite fields.  In particular, we showed

Kakeya maximal function conjecture in algebraic varieties. Suppose that $W \subset {\Bbb P}^N$ is an (n-1)-dimensional algebraic variety.  Let $d \geq 1$ be an integer. Then we have

$\| \sup_{\gamma \ni x; \gamma \not \subset W} \sum_{y \in \gamma} f(y) \|_{\ell^n_x(W(F))} \leq C_{n,d,N,W} |F|^{(n-1)/n} \|f\|_{\ell^n({\Bbb P}^N(F))}$

for some constant $C_{n,d,N,W} > 0$, where the supremum is over all irreducible algebraic curves $\gamma$ of degree at most d that pass through x but do not lie in W, and W(F) denotes the F-points of W.

The ordinary Kakeya maximal function conjecture corresponds to the case when N=n, W is the hyperplane at infinity, and the degree d is equal to 1.  One corollary of this estimate is a Dvir-type result: a subset of ${\Bbb P}^N(F)$ which contains, for each x in W, an irreducible algebraic curve of degree d passing through x but not lying in W, has cardinality $\gg |F|^n$ if $|W| \gg |F|^{n-1}$.  (In particular this implies a lower bound for Nikodym sets worked out by Li.)  The dependence of the implied constant on W is only via the degree of W.

The techniques used in the flat case can easily handle curves $\gamma$ of higher degree (provided that we allow the implied constants to depend on d), but the method of random rotations does not seem to work directly on the algebraic variety W as there are usually no symmetries of this variety to exploit.  Fortunately, we can get around this by using a “random projection trick” to “flatten” W into a hyperplane (after first expressing W as the zero locus of some polynomials, and then composing with the graphing map for such polynomials), reducing the non-flat case to the flat case.

Below the fold, I wish to sketch two of the key ingredients in our arguments, the random rotations method and the random projections trick.  (We of course also use some algebraic geometry, but mostly low-tech stuff, on the level of Bezout’s theorem, though we do need one non-trivial result of Kleiman (from SGA6), that asserts that bounded degree varieties can be cut out by a bounded number of polynomials of bounded degree.)

[Update, March 14: See also Jordan's own blog post on our paper.]