Curved Kakeya sets for generic phases in odd dimensions

TL;DR

This paper demonstrates that for odd dimensions, generic H"ormander phase functions produce curved Kakeya sets with Hausdorff dimension exceeding classical thresholds, using polynomial methods and generalizations of contact order conditions.

Contribution

It extends the finite contact order condition to all odd dimensions and derives stronger curved Kakeya estimates directly via polynomial methods.

Findings

Curved Kakeya sets in odd dimensions have Hausdorff dimension at least (n+1)/2 + d_n.

In R^3, generic phases produce sets of dimension at least 2 + 1/7.

Quantitative improvements in oscillatory integral bounds for positive-definite phases in all odd dimensions.

Abstract

We show that for each odd integer $n\ge 3$, there is an open dense subset of H\"ormander phase functions in $\mathbb{R}^n$ for which the associated curved Kakeya sets have Hausdorff dimension at least $\frac{n+1}{2} + d_n$ for some positive $d_n$, thereby exceeding the classical compression threshold. In particular, in $\mathbb{R}^3$, generic H\"ormander phases induce curved Kakeya sets of dimension at least $2 + \tfrac17$. As an application, on a generic three-dimensional Riemannian manifold, a local Nikodym set has Hausdorff dimension at least $2 + \tfrac17$. We achieve these results by generalizing the finite contact order condition from Dai--Gong--Guo--Zhang, originally developed in $\mathbb{R}^3$, to arbitrary dimensions. Our bounds are stronger than those of Dai--Gong--Guo--Zhang even in $\mathbb{R}^3$, since we derive curved Kakeya estimates directly via the polynomial method. Moreover, for H\"ormander-type oscillatory integral operators with positive-definite phases of finite contact order, we obtain quantitative improvements in all odd dimensions over the bounds of Guth--Hickman--Iliopoulou, while in three dimensions our oscillatory integral estimate exactly matches the result of Dai--Gong--Guo--Zhang.