Improved packing of hypersurfaces in $\mathbb R^d$

TL;DR

The paper addresses packing curved hypersurfaces with nonzero Gaussian curvature into a thin subset of $\mathbb{R}^{d+1}$ by constructing a compact set whose $\delta$-neighborhood has measure decaying like $|\log\delta|^{-2/d}$. It develops a two-tier approach: first, packing graphs of regular curved functions via carefully controlled translations along a grid-path, producing a small $2^{-M^d}$-scale thickness; then, extending to general curved hypersurfaces through a partition-of-unity and iterative translations to achieve global thinness. A key contribution is the removal of a logarithmic log-log factor relative to earlier work, yielding sharp scaling for $d=2$ and robust bounds for higher $d$, including Hölder regularity $C^{2,\alpha}$. The results illuminate the role of curvature in Kakeya-type packing, showing that nonzero Gaussian curvature enables significantly thinner coverings than straight-line analogues and providing a framework that applies to a broad class of hypersurface families.

Abstract

For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $δ\in (0,1)$, the $δ$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log δ|^{-2/d}$. This is the smallest possible order when $d=2$, and improves a result of Kolasa-Wolff (Pacific J. Math., 190(1):111-154, 1999). Our construction also generalises to Holder-continuous families of $C^{2,α}$ hypersurfaces with nonzero Gaussian curvature.

Figures (1)

• Figure 1: Translations of the annuli after 3 steps

Theorems & Definitions (11)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof