Generalized Besicovitch-Eggleston Function And Self-similar Structure Of Cartesian Product Of Cantor Sets

In the present paper, we study two kinds of fractal sets. The first one is the graph of the Besicovitch-Eggleston vector function. This function is defined by the frequencies of digits, occurring in the N-adic expansion of the variable and its graph is a subset of [0,1]N with complicated geometric structure and zero-value topological dimension. We explore its properties and show it has Hausdorff dimension N. The second one is related to the Cartesian product of the middle-(1-2β) Cantor sets. Such sets are a natural generalization of the classical middle-third Cantor sets. Compared with the latter they are constructed by removing middle-(1-2β) intervals on each step, instead of one third. We obtain the sufficient and necessary conditions for the intersection of this product with its translation to be a self-similar set.