Point-extended box dimension

Abstract

11We would like to extend our thanks to the College of Engineering and Architecture of the International University of Rabat for supporting this project through one-year teaching discharge granted to professor Maaroufi Nadir to finalize the drafting of this article. This article is mainly concerned by the concept of dimension. More precisely, our objective is to challenge the conventional zero dimension assigned to the point. This reconsideration allows us to propose two new ways of conceiving the notion of dimension, which are the two sides of the same coin. First, the dimension of a set appears as a quantification of the organization of its points. Secondly, the dimension seems essentially to be a comparison between the entropies of sets. Thus, we started from the point and succeeded in constructing a point-dimension notion allowing us to extend the principle of box dimension in many directions and to finely estimate the dimensions of sets. More precisely, we introduce the notion of point-extended box dimension in the large framework of topological vector spaces, freeing it from the notion of metric. This general setting permits us to treat the case of finite, infinite and invisible dimensions. This first work focuses essentially on general properties and is particularly oriented towards establishing a well founded framework for infinite dimension.