Directional maximal operators and lacunarity in higher dimensions

We introduce a notion of lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p({\Bbb R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel-Stein-Wainger, Sj\"ogren-Sj\"olin and Carbery. Closely related to this, we find a characterization of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue-type differentiation of integrals in $L_{{\rm loc}}^p({\Bbb R}^n)$, replacing balls by tubes which point in these directions.