Kakeya Sets in Cantor directions

2006/09/06 by Michael Bateman, Bateman, Michael D., Nets Hawk Katz +1 · 1 citation
Mathematics · #42B25 #60K35 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/0609187

openalex publication_date 2006/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L² 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.

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