New curved Kakeya estimates

TL;DR

This work advances curved Kakeya estimates in $\mathbb{R}^3$ by marrying Wolff’s hairbrush method with a novel incidence bound for 3-parameter curve families that satisfy coniness and twistiness. By analyzing translation-invariant, negatively curved phase functions in normal form and leveraging a refined 3-parameter Kakeya framework, the author proves a new lower bound for Kakeya-type sets: $d_{\mathrm{set}}(\phi) \geq \frac{13-\sqrt{13}}{4} \approx 2.348$, surpassing the previous $2+\tfrac{1}{3}$ barrier in many curved settings. A model class $\phi_A$ is studied to relate Kakeya compression to the exponent via $d_{\max}(\phi_A) \leq 2+\tfrac{1}{m(A)}$, while a comprehensive treatment of 3-parameter families introduces notions of coniness and twistiness, grain structure, and prism decompositions. The results connect curved Kakeya to nonlinear restricted projections, yielding corollaries for dimension bounds of nonlinear projections and offering a flexible framework that generalizes KWZ in the curved, variable-coefficient regime. Overall, the paper provides a robust incidence-based approach for curved Kakeya in 3D, opening pathways to sharper bounds and broader applications in harmonic analysis and nonlinear projection problems.

Abstract

We give new lower bounds for the Hausdorff dimension of Kakeya sets built from various families of curves in $\mathbb R^3$, going beyond what the polynomial partitioning method has so-far achieved. We do this by combining Wolff's classical hairbrush argument with a new incidence bound for 3-parameter families of curves which satisfy conditions we call coniness and twistiness. Our main argument builds off a technique of Katz, Wu, and Zahl used in the study of $\rm{SL}_2$-Kakeya sets.

Figures (5)

• Figure 1: The union of cubes $E$ (in black). There are 4 curves in $L$ depicted (in blue). The shading density in the picture is $\lambda \approx 6 \cdot\delta$.
• Figure 2: Taxonomy of phase functions based on the strength of polynomial Wolff axioms. From bottom to top, the available polynomial Wolff axioms tend to get stronger, and the counterexamples to Problems \ref{['prob: hormander']} and \ref{['prob: curved kakeya maximal']} tend to get weaker. This is what is meant by "Better Behaved." The gray strip on the left is a cross section in a special model case. The dimensions $d_{\rm{set}}(\phi)$ and $d_{\rm{max}}(\phi)$ jumping "out of the page" for this cross section are in Figure \ref{['fig:toy_case_diagram']}, which is discussed in Subsection \ref{['subsec: special case']}. From left to right, one should think of the examples as becoming more "complicated." At the bottom level, we have the phase functions giving rise to Kakeya sets that lie in a surface, such as $\phi_{\rm{worst}}$BourgainSeveralVariables, the Minocozzi-Sogge example minicozziSogge, and the "$\log$"-example beyondUniversalEstimates. The breadth of examples at this level are not yet totally understood. The "Kakeya noncompression" level guarantees that a Kakeya set cannot compress into a surface, but it is preceded by "?," since it is not clear that this is the minimal condition. Above this is the hierarchy of contact orders from DaiOscillatory. Above this is another "?," since Contact order $\leq k$ for $k < 4$ is not defined. In Subsection \ref{['subsec: special case']} we will see that our main result lies in this region, at least in the model case. At the top is Bourgain's condition, which includes $\phi_{\rm{parab}}$ and constant curvature reduced Carleson-Sjölin. It also includes $\phi_{\tan}$, among other new examples which are less understood. The main progress at this top level is from HormanderDichotomy.
• Figure 3: Plot of the ranges of $d_{\rm{max}}(\phi_A)$ (blue) and $d_{\rm{set}}$ (red), depending on the level of compression (measured by $m(A)$ and $m^*(A)$). The "$1$"-endpoint contains those phases with robust compression $1$, as explained after Remark \ref{['rmk: wisewell']}. This is classical Kakeya. The "2"-bracket is $\{\phi_A : m(A) = 2\}$, where Theorem \ref{['thm: main phicurved kakeya']} applies (highlighted in green). This region corresponds to the gray strip in the "?"-region between contact order $\leq 4$ and the Bourgain condition in Figure \ref{['fig: schematicOfCurvedKakeya']}. For $m \geq 3$, the "$m$"-bracket is $\{\phi_A : m(A) = m^*(A) = m\}$ (where $m^*(A) = m$ is required for contact order to apply). The positive results here are \ref{['eq: contact order kakeya']}, based on DaiOscillatory. At the very right is the "$\infty$"-endpoint, containing the examples $\{\phi_A : m(A) = \infty\}$. The trivial bounds are sharp here. The upper bound on $d_{\rm{max}}$ is Proposition \ref{['prop: max fnc upper bound']}.
• Figure 4: Prism $P$ of dimensions $\delta \times \rho \times d$ around $\ell$. The green shadings $Y(\ell')$ intersect $Y(\ell)$ at angle $\sim r$, and fill out $P$.
• Figure 5: The grain structure of $\mathcal{C}(X)$ inside $N_\rho(\ell_0)$. Another curve $\ell' \in B(\ell_0, \rho)$ is depicted in red. The prisms in $\mathcal{P}_\rho(\ell_0)$ intersecting $\ell_0$ and $\ell'$ are colored light blue and light red respectively, and two such prisms are labeled $P$ and $P'$ respectively. To the right, we show a cross section of $N_\rho(\ell_0)$. The cross sections of the grains are parallel up to error $\delta$, and pointing in direction $\gamma_{\mathbf p_0}(t)$. At the bottom is the projection of the grains in $N_\rho(\ell_0)$ under $\pi_{\mathbf p_0}$. The light-blue grains project to the blue line, and the light-red grains project to the red curve. This captures the geometry used in Lemma \ref{['lem: add prisms fat tube']}.

Theorems & Definitions (118)

• Definition 1.1
• Definition 1.2
• Conjecture 1.3: Restriction Conjecture
• Definition 1.4: $\phi$-curves
• Example 1.5: Straight lines
• Conjecture 1.6: Kakeya Maximal Function Conjecture
• Definition 1.7: $\phi$-curved Kakeya Set
• Conjecture 1.8: Kakeya Set Conjecture
• Definition 1.9: Bourgain's Condition
• Definition 1.10: Polynomial Wolff Axioms for $\phi$