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arXiv · 2503.15760

New curved Kakeya estimates

Arian Nadjimzadah
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Abstract

We prove that for an open class of translation-invariant H\"ormander-type phase functions, the corresponding curved Kakeya sets in $\mathbb R^3$ have Hausdorff dimension at least $(13-\sqrt{13})/4 = 2.348\ldots$. Previous methods, such as polynomial partitioning, have attained at best dimension $2+1/3$ for curved Kakeya sets in $\mathbb R^3$. The proof combines Wolff's classical hairbrush argument with a new Kakeya estimate for 3-parameter families of curves satisfying stable geometric conditions that we call coniness and twistiness. The latter estimate extends ideas of Katz, Wu, and Zahl from the study of $\mathrm{SL}_2$-Kakeya sets.

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Arian Nadjimzadah. 2025-03-20. New curved Kakeya estimates. https://arxiv.org/abs/2503.15760

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