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Construction of a curved Kakeya set

Tongou Yang, Yue Zhong

TL;DR

This work constructs a zero-measure curved Kakeya set in the plane that contains a piece of a parabola for every aperture in $[1,2]$, and refines lower bounds for the associated maximal operators. The method hinges on a cut-and-slide compression that forces tangencies among curved rectangles, producing thickened sets with tightly controlled measure. A generalization to graphs of $C^2$ functions under a curvature condition shows the approach applies beyond parabolas, yielding sharp lower bounds $R(p,q,\delta)\gtrsim (\log \delta^{-1})^{2/p}$ and extending the zero-measure construction via an iterative Cantor-like scheme. The results sharpen understanding of $L^p$-$L^q$ bounds for curved maximal operators under cinematic curvature and provide a framework for further refinements and open questions in this area.

Abstract

We construct a compact set in $\mathbb R^2$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a range of $p$, $q$. Moreover, our construction can be generalised from parabolas to a family of $C^2$ curves satisfying suitable curvature conditions.

Construction of a curved Kakeya set

TL;DR

This work constructs a zero-measure curved Kakeya set in the plane that contains a piece of a parabola for every aperture in , and refines lower bounds for the associated maximal operators. The method hinges on a cut-and-slide compression that forces tangencies among curved rectangles, producing thickened sets with tightly controlled measure. A generalization to graphs of functions under a curvature condition shows the approach applies beyond parabolas, yielding sharp lower bounds and extending the zero-measure construction via an iterative Cantor-like scheme. The results sharpen understanding of - bounds for curved maximal operators under cinematic curvature and provide a framework for further refinements and open questions in this area.

Abstract

We construct a compact set in of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the - norm of the corresponding maximal operator for a range of , . Moreover, our construction can be generalised from parabolas to a family of curves satisfying suitable curvature conditions.
Paper Structure (21 sections, 9 theorems, 79 equations, 2 figures, 1 table)

This paper contains 21 sections, 9 theorems, 79 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Curves of cinematic curvature
  3. Lower bounds
  4. A generalised curved Kakeya set
  5. Outline of the article
  6. Acknowledgements
  7. Compression by forcing tangencies
  8. Step 0
  9. Step 1
  10. Step $2$
  11. Step $j$
  12. End of construction
  13. Computation of translations
  14. Upper bound of measure
  15. Construction of zero measure Kakeya set
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a compact subset $K$ of $\mathbb{R}^2$ of Lebesgue measure $0$ that contains a piece of length $\sim 1$ of a parabola of every aperture between $1$ and $2$. Moreover, its $\delta$-thickening $K(\delta)$ satisfies $|K(\delta)|\lesssim (\log \delta^{-1})^{-2}(\log \log \delta^{-1})^{2}$. where the implicit constant is independent of $\delta$. In particular, $B(p,q)=\infty$ if $p<\infty

Figures (2)

  • Figure 1: Figure after two steps when $2^M=16$.
  • Figure 2: Interpolation diagram of $\mathcal{R}_\delta$

Theorems & Definitions (15)

  • Theorem 1.1: Parabolic variant of Kolasa-Wolff construction Kolasa_Wolff
  • Theorem 1.2: Main theorem
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • proof : Proof of corollary assuming Theorem \ref{['thm_measure_bound']}
  • ...and 5 more