The authors pack every d-sphere of radii 1 to 2 into a set whose δ-neighborhood has measure ≲ |log δ|^{-2/d}, matching the lower bound in two dimensions.
Construction of a curved Kakeya set
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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.
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math.CA 1years
2025 1verdicts
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Improved packing of hypersurfaces in $\mathbb R^d$
The authors pack every d-sphere of radii 1 to 2 into a set whose δ-neighborhood has measure ≲ |log δ|^{-2/d}, matching the lower bound in two dimensions.