[Paper Review] Kakeya Sets in Cantor directions

Michael Bateman, Nets Hawk Katz|ArXiv.org|Sep 6, 2006

Mathematical Dynamics and Fractals5 references4 citations

TL;DR

This paper constructs a family of Kakeya sets in the plane using parallelograms with slopes in the standard middle-thirds Cantor set, demonstrating that such sets can have arbitrarily small Lebesgue measure (on the order of $1/\log N$) despite containing line segments in a uncountable set of directions. The construction relies on a probabilistic method and electrical network theory on trees, yielding a maximal operator that is unbounded on all $L^p$ spaces for $p \neq \infty$, resolving a long-standing open problem in harmonic analysis.

ABSTRACT

We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of the doubles has area log log N/ log N. In particular, this implies unbounded of the associated maximal operator in L^p for any p different from infinity. The construction is by randomizing an earlier construction of the second author for the L^2 case. The proof that the construction satisfies the desired conditions is by elementary estimates in the theory of percolation on trees as developed by R. Lyons.

Motivation & Objective

• To resolve the long-standing open problem of the unboundedness of the maximal operator $\mathcal{M}$ associated with line segments in directions from the standard Cantor set on $L^p(\mathbb{R}^2)$ for $p \neq \infty$.
• To construct explicit examples of Kakeya-type sets with slopes restricted to the Cantor set but with arbitrarily small Lebesgue measure.
• To establish a connection between percolation on trees and the measure-theoretic properties of Kakeya sets in fractal directions.
• To provide a probabilistic construction of a Kakeya set whose measure decays logarithmically with $N$, while its dilated version has measure bounded below by $\log \log N / \log N$.

Proposed method

• A probabilistic construction of sticky maps $\sigma_n: T_n \to C_n$, where $T_n$ is a ternary tree and $C_n$ is the $n$-stage Cantor set, using independent random variables $r_{t,a} \in \{0,2\}$ to define slopes.
• Definition of parallelograms $P_{\sigma,s}$ with eccentricity $\sim N = 3^n$ and area $\sim 1/N$, aligned along directions in the Cantor set.
• Use of electrical network resistance theory on subtrees of the ternary tree to estimate the probability that a point lies in the union of parallelograms.
• Application of Lyons' theorem relating the probability of hitting a subtree to the effective resistance $R(T')$, with $R(T') \gtrsim n$ for relevant subtrees.
• Use of a uniformity inequality in measure theory to lower-bound the measure of the union of parallelograms based on pairwise intersections.
• Interchange of integrals and expectation to compute the expected measure of the Kakeya set, leading to the key decay estimate.

Experimental results

Research questions

• RQ1 Can a Kakeya set be constructed with directions restricted to the Cantor set and still have arbitrarily small Lebesgue measure?
• RQ2 Is the maximal operator $\mathcal{M}f(x) = \sup_{x \in s \in \mathcal{S}} \text{av}_s |f|$ unbounded on $L^p(\mathbb{R}^2)$ for $p \neq \infty$, where $\mathcal{S}$ is the set of lines with slopes in the Cantor set?
• RQ3 What is the asymptotic behavior of the measure of a Kakeya set formed by $N = 3^n$ parallelograms of eccentricity $\sim N$ and area $\sim 1/N$ with slopes in the Cantor set?
• RQ4 How does the measure of the dilated set $2P_j$ compare to the original set in such constructions?

Key findings

• For $N = 3^n$, there exists a union of $N$ parallelograms with slopes in the Cantor set, each of eccentricity $\sim N$ and area $\sim 1/N$, such that the measure of their union satisfies $|\bigcup_{j=1}^N P_j| \lesssim 1/\log N$.
• The measure of the dilated union $\bigcup_{j=1}^N 2P_j$ satisfies $|\bigcup_{j=1}^N 2P_j| \gtrsim \log \log N / \log N$.
• The maximal operator $\mathcal{M}$ defined over lines with slopes in the standard Cantor set is unbounded on $L^p(\mathbb{R}^2)$ for all $p \neq \infty$.
• The construction relies on a probabilistic model of sticky maps on a ternary tree, with the probability that a point lies in the Kakeya set estimated via effective resistance in an electrical network.
• The key lower bound on the measure of the Kakeya set is $|K_\sigma| \gtrsim \log n / n$, which is derived from the resistance estimate $R(T') \gtrsim n$ and Lyons' theorem.
• The expected measure of the Kakeya set over the probability space of sticky maps is $\lesssim 1/n$, implying the existence of a specific instance with measure $\lesssim 1/n$, while the lower bound $\gtrsim \log n / n$ ensures non-triviality.