Exceptional set estimate in prime fields: dimension two implies higher dimensions

Exceptional set estimate in prime fields: dimension two implies higher dimensions, Discrete Analysis 2025:19, 55 pp.

Let \(\mathcal{L}\) be a family of lines in \(\mathbb R^2\). We say that \(\mathcal{L}\) is \(t\)-dimensional if \(t\) is the Hausdorff dimension of the set of pairs \((a,b)\in\mathbb R^2\) such that the line \(y=ax+b\) belongs to \(\mathcal{L}\). (For ease of exposition, we assume that all lines in \(\mathcal{L}\) are non-vertical, but this restriction is easy to remove.) A set \(F\subset\mathbb R^2\) is called an \((s,t)\)-Furstenberg set if there is a \(t\)-dimensional family of lines \(\mathcal{L}\) such that \(F\) intersects every line \(L\in\mathcal{L}\) in a set of Hausdorff dimension at least \(s\). What can we say about lower bounds on the Hausdorff dimension of \((s,t)\)-Furstenberg sets? This question was posed by Wolff in connection with his work on the Kakeya problem, which, in the plane, corresponds to letting \(t=s=1\). It was resolved recently (in 2023) by Orponen-Shmerkin and Ren-Wang who proved optimal dimension bounds for Furstenberg sets in the plane (the two papers considered different cases of the problem).

There are several natural directions in which the Furstenberg problem can be generalized. One is that, instead of working in the plane, one may consider sets in \(\mathbb R^n\) with \(n\geq 3\) that have large intersections with families of affine subspaces of some dimension \(1\leq k\leq n-1\). This higher-dimensional question remains open, although several partial results are available.

In a different direction, an analogous problem (in dimension 2 or higher) may be considered over finite fields. The finite field Kakeya problem was first introduced by Wolff for expository purposes. In 2008, Dvir gave a simple and elegant solution of that problem in all dimensions; however, attempts to replicate his solution in the Euclidean case were not successful. It was only earlier this year (2025) that the 3-dimensional Kakeya conjecture was proved by Wang and Zahl; the higher-dimensional case remains open.

Based on the above, one might expect that the finite field Furstenberg problem should be easier than the Euclidean version. That, however, turns out not to be the case. Despite the Ren-Wang and Orponen-Shmerkin breakthroughs in the plane, the finite field 2-dimensional case remains open, although here, too, there are partial results.

The present article by Shengwen Gan studies the finite field Furstenberg problem. The main result is that, somewhat surprisingly, and unlike in the Euclidean case, the optimal 2-dimensional estimate in finite fields would also imply the optimal \(n\)-dimensional estimate. This in an interesting and unexpected result. The author gives a reduction for the lower bounds, as well as matching constructions of small Furstenberg sets that show these lower bounds to be optimal. The proof is based on exceptional set estimates for projection estimates, likely to be of interest in their own right.