Curved Kakeya sets for generic phases in odd dimensions

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.