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We develop a notion of finite order lacunarity for direction sets in $\mathbbR^{d+1}$. Given a direction set $\Omega$ that is sublacunary according to thisdefinition, we construct random examples of Euclidean sets that contain unitline segments with directions from $\Omega$ and enjoy analytical featuressimilar to those of traditional Kakeya sets of infinitesimal Lebesgue measure.This generalizes to higher dimensions a planar result due to Bateman. Combinedwith earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion oflacunarity and Kakeya-type sets also yields a characterization in alldimensions for directional maximal operators to be $L^p$-bounded.

Lacunarity, Kakeya-type sets and directional maximal operators