New bounds on Cantor maximal operators

. We prove L p bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Laba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors- regular Salem Cantor sets of any dimension > 1 / 2 such that the associated maximal operator is bounded on L 2 ( R ). We follow the overall scheme of Laba– Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures.