BOX-COUNTING DIMENSION COMPUTED BY α-DENSE CURVES

Abstract

We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube [Formula: see text], [Formula: see text]. The procedure is based on the existence of special types of [Formula: see text]-dense curves (a generalization of the space-filling curves) in [Formula: see text] called [Formula: see text]-uniform curves.