弯曲Kakeya问题与路径的射影几何

中文摘要 AI 辅助

我们为$\boldsymbol{R}^n$中的弯曲Kakeya问题引入了一个通用框架，涵盖由Hörmander型振荡积分产生的相关问题。每类曲线族都确定一种喷雾几何，这使我们能利用路径的射影几何来研究弯曲Kakeya问题。我们聚焦弯曲Kakeya集可能呈现的“最优”与“最差”两种极端行为，刻画Wolff毛刷论证所基于的关联结构何时成立。特别地，我们证明Wolff毛刷论证所需的大量全测地曲面的存在性，等价于关联喷雾的射影平坦性。在该射影平坦类中，Bourgain条件给出清晰二分：若该条件成立，则此曲线族在方向上等价于Bochner–Riesz型直线族，且满足Katz–Wolff条件，从而可应用Wang–Zahl结果；若该条件不成立，则每个全测地曲面上都承载一个二维Kakeya集。我们还证明，在额外的半代数假设下，$\boldsymbol{R}^3$中的曲线族存在Hausdorff维数为2的弯曲Kakeya集，当且仅当它存在含于某曲面内的弯曲Kakeya集；等价地，若不存在这种压缩，则每个关联弯曲Kakeya集的维数严格大于2。

英文摘要

We introduce a general framework for curved Kakeya problems in $\mathbb{R}^n$, encompassing those arising from Hörmander-type oscillatory integrals. Every family of curves determines a spray geometry, which allows us to use the projective geometry of paths in the study of curved Kakeya problems. We focus on the two extremes of the "best" and "worst" possible behaviors of curved Kakeya sets. We characterize when the incidence structure underlying Wolff's hairbrush argument persists. In particular, we prove that the existence of many totally geodesic surfaces, as required by Wolff's hairbrush argument, is equivalent to projective flatness of the associated spray. Within this projectively flat class, Bourgain's condition provides a clean dichotomy: when it holds, the family is direction-equivalent to a Bochner--Riesz type family of lines and satisfies the Katz--Wolff condition, and thus the Wang--Zahl result is applicable; when it fails, every totally geodesic surface supports a two-dimensional Kakeya set. We also show that under an extra semi-algebraic assumption, a family of curves in $\mathbb{R}^3$ admits a curved Kakeya set of Hausdorff dimension $2$ if and only if it admits a curved Kakeya set contained in a surface. Equivalently, if this compression is absent, every associated curved Kakeya set has dimension strictly greater than $2$.

补充信息

备注

90 pages, 2 figures. Comments welcome!