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Curved Kakeya sets and Nikodym problems on manifolds

Chuanwei Gao, Diankun Liu, Yakun Xi

TL;DR

This work links curved Kakeya and Nikodym problems on manifolds to their Euclidean counterparts by straightening geodesics via local diffeomorphisms. It shows that translation-invariant phase functions satisfying Bourgain's condition can be transformed to a standard linear form, enabling the transfer of all known Kakeya bounds and maximal-function estimates to curved settings. Consequently, Nikodym-type results on space forms with constant sectional curvature reduce to Euclidean Kakeya problems, yielding the Nikodym conjecture in dimension three through the Wang–Zahl breakthrough. The paper also extends these ideas to $(d,k)$-Nikodym and $(s,t)$-Furstenberg sets on manifolds and surfaces, establishing dimension bounds that mirror Euclidean results and unifying several geometric harmonic analysis problems under the curved-Kakeya framework.

Abstract

In this paper, we study curved Kakeya sets associated with phase functions satisfying Bourgain's condition. In particular, we show that the analysis of curved Kakeya sets arising from translation-invariant phase functions under Bourgain's condition, as well as Nikodym sets on manifolds with constant sectional curvature, can be reduced to the study of standard Kakeya sets in Euclidean space. Combined with the recent breakthrough of Wang and Zahl, our work establishes the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature. Moreover, we consider $(d,k)$-Nikodym sets and $(s,t)$-Furstenberg sets on Riemannian manifolds. For manifolds with constant sectional curvature, we prove that these problems can similarly be reduced to their Euclidean counterparts. As a result, the Furstenberg conjecture on two-dimensional surfaces with constant Gaussian curvature follows from the work of Ren and Wang.

Curved Kakeya sets and Nikodym problems on manifolds

TL;DR

This work links curved Kakeya and Nikodym problems on manifolds to their Euclidean counterparts by straightening geodesics via local diffeomorphisms. It shows that translation-invariant phase functions satisfying Bourgain's condition can be transformed to a standard linear form, enabling the transfer of all known Kakeya bounds and maximal-function estimates to curved settings. Consequently, Nikodym-type results on space forms with constant sectional curvature reduce to Euclidean Kakeya problems, yielding the Nikodym conjecture in dimension three through the Wang–Zahl breakthrough. The paper also extends these ideas to -Nikodym and -Furstenberg sets on manifolds and surfaces, establishing dimension bounds that mirror Euclidean results and unifying several geometric harmonic analysis problems under the curved-Kakeya framework.

Abstract

In this paper, we study curved Kakeya sets associated with phase functions satisfying Bourgain's condition. In particular, we show that the analysis of curved Kakeya sets arising from translation-invariant phase functions under Bourgain's condition, as well as Nikodym sets on manifolds with constant sectional curvature, can be reduced to the study of standard Kakeya sets in Euclidean space. Combined with the recent breakthrough of Wang and Zahl, our work establishes the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature. Moreover, we consider -Nikodym sets and -Furstenberg sets on Riemannian manifolds. For manifolds with constant sectional curvature, we prove that these problems can similarly be reduced to their Euclidean counterparts. As a result, the Furstenberg conjecture on two-dimensional surfaces with constant Gaussian curvature follows from the work of Ren and Wang.

Paper Structure

This paper contains 14 sections, 18 theorems, 93 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kakeya sets, Nikodym sets and curved Kakeya sets
  3. Other Nikodym-type sets on manifolds
  4. Organization
  5. Acknowledgements.
  6. The Nikodym problem on manifolds
  7. Nikodym sets on manifolds
  8. The Nikodym maximal function on manifolds
  9. Nikodym-type sets on manifolds
  10. $(d,k)$-Nikodym sets on manifolds
  11. $(s,t)$-Furstenberg sets on Riemannian surfaces
  12. Translation-invariant phases under Bourgain's condition
  13. Proof of Theorem \ref{['theomain']}
  14. Applications

Key Result

Theorem 1.5

Suppose that in $\mathbb{R}^d$, any Kakeya set must have Hausdorff (resp. Minkowski) dimension at least $\alpha$. Then any Nikodym set on a manifold $M^d$ with constant sectional curvature must also have Hausdorff (resp. Minkowski) dimension at least $\alpha$. In particular, the Kakeya conjecture in

Figures (3)

  • Figure 1: The straightening map on the sphere
  • Figure 2: The straightening map on the hyperbolic space
  • Figure 3: Geodesics in the Beltrami-Klein model

Theorems & Definitions (50)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7: Hörmander's non-degeneracy condition
  • Definition 1.8: Kakeya Curves
  • Definition 1.9: Curved Kakeya sets
  • Definition 1.10: Translation-invariant condition Bourgain91CGGHIW24
  • ...and 40 more