标题： 奇数维中通用相位的曲线Kakeya集 显示英文标题

标题： Curved Kakeya sets for generic phases in odd dimensions

摘要： 我们证明对于每个奇数整数$n\ge 3$，在$\mathbb{R}^n$中存在一个开稠密子集的 Hörmander 相位函数，使得相关的弯曲 Kakeya 集的豪斯多夫维数至少为$\frac{n+1}{2} + d_n$，对于某些正数$d_n$，从而超过经典的压缩阈值。特别地，在$\mathbb{R}^3$中，一般的 Hörmander 相位函数会诱导维数至少为$2 + \tfrac17$的弯曲 Kakeya 集。作为应用，在一般的三维黎曼流形上，每个测地线 Nikodym 集的豪斯多夫维数至少为$2 + \tfrac17$。 我们通过将Dai--Gong--Guo--Zhang在$\mathbb{R}^3$中最初开发的有限接触阶条件推广到任意维度，从而获得了这些结果。 我们的界限在$\mathbb{R}^3$中甚至比Dai--Gong--Guo--Zhang的界限更强，因为我们通过多项式方法直接得出了弯曲Kakeya估计。 此外，对于具有有限接触阶正定相位的Hörmander型振荡积分算子，我们在所有奇数维上都获得了Guth--Hickman--Iliopoulou界限的定量改进，而在三维情况下，我们的振荡积分估计恰好与Dai--Gong--Guo--Zhang的结果一致。 显示英文摘要

摘要： 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, every geodesic 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.