  The criterion of the recurrent compact set was introduced by Moreira and
Yoccoz to prove that stable intersections of regular Cantor sets on the real
line are dense in the region where the sum of their Hausdorff dimensions is
bigger than 1. We adapt this concept to the context of horseshoes in ambient
dimension higher than 2 and prove that horseshoes with upper stable dimension
bigger than 1 satisfy, typically and persistently, the adapted criterion of the
recurrent compact set. As consequences we show some persistent geometric
properties of these horseshoes. In particular, typically and persistently,
horseshoes with upper stable dimension bigger than 1 present blenders.
