Curved Kakeya sets for generic phases in odd dimensions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918141025583104 |
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| author | Guo, Shaoming Liu, Diankun Xi, Yakun |
| author_facet | Guo, Shaoming Liu, Diankun Xi, Yakun |
| contents | 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 $2 + \tfrac17$. 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17706 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curved Kakeya sets for generic phases in odd dimensions Guo, Shaoming Liu, Diankun Xi, Yakun Classical Analysis and ODEs 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 $2 + \tfrac17$. 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 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. |
| title | Curved Kakeya sets for generic phases in odd dimensions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.17706 |