Curved Kakeya sets for generic phases in odd dimensions

Abstract

We show that for each odd integer $n\ge 3$, there is an open dense subset of Hörmander 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 Hörmander phases induce curved Kakeya sets of dimension at least $2+\frac{1}{7}$. As an application, on a generic three-dimensional Riemannian manifold, a local Nikodym set has Hausdorff dimension at least $2+\frac{1}{7}$. 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 Hörmander-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.

37 pages. Minor inaccuracies in the statement of our results have been addressed