Box-counting dimension computed by α-dense curves

We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube In, n > 1. The procedure is based on the existence of special types of α-dense curves (a generalization of the space-filling curves) in In called δ-uniform curves.This work was partially supported by the grant from the Ministerio de Economía y Competitividad, Spain (MTM 2014-52865-P) to the second author