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Abstract

Given a Cantor-type subset Ω of a smooth curve in ℝ(d+1), we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in ℝ(d+1), and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz [4], [3]. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(ℝ(d+1)) for all 1 ≤ p < ∞.