Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem

TL;DR

This paper proves discrete analogues of the Kakeya conjecture and Furstenberg set problem, showing that low-dimensional Kakeya sets cannot be formed from discrete configurations, using polynomial methods and zero set density arguments.

Contribution

It establishes the first discrete versions of these conjectures, generalizes the Furstenberg problem, and introduces new polynomial techniques and conjectures related to zero set density.

Findings

Discrete Kakeya conjecture is proven in all dimensions.

Discrete Furstenberg set problem is completely solved.

A new polynomial-based approach to zero set density is developed.

Abstract

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The difference between our theorems and the (true) problems is only the (still difficult) issue of continuity since no transversality-at-incidences assumptions are imposed. The main tool of the proof is a theorem of Wongkew \cite{wongkew2003volumes} which states that a low degree polynomial cannot have its zero set being too dense inside the unit cube, coupled with Dvir-type polynomial arguments \cite{dvir2009size}. From the viewpoint of the proofs, we also state a conjecture that is stronger than and almost equivalent to the (lower) Minkowski version of the Kakeya conjecture and prove some results towards it. We also present our own version of the proof of the theorem in \cite{wongkew2003volumes}. Our proof shows that this theorem follows from a combination of properties of zero sets of polynomials and a general proposition about hypersurfaces which might be of independent interest. Finally, we discuss how to generalize Bourgain's conjecture to high dimensions, which is closely related to the theme here.