Curved Kakeya sets and Nikodym problems on manifolds

Paper

Curved Kakeya sets and Nikodym problems on manifolds

Published Mar 14, 2025 · Chuanwei Gao, Diankun Liu, Yakun Xi

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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.

Study Snapshot

Curved Kakeya sets and Nikodym sets on manifolds with constant sectional curvature can be reduced to standard Kakeya sets in Euclidean space, establishing the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature.

PopulationOlder adults (50-71 years)

Sample size24

MethodsObservational

OutcomesBody Mass Index projections

ResultsSocial networks mitigate obesity in older groups.

Curved Kakeya sets and Nikodym problems on manifolds

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