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

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 main tool of the proof is a theorem of Wongkew (Pac J Math 159:177–184, 2003), 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 (Dvir in J Am Math Soc 22(4):1093–1097, 2009). 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 toward it. We also present our own version of the proof of the theorem in Wongkew (Pac J Math 159:177–184, 2003). 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, along the way providing a counterexample to the most naive generalization.