EDB — 0Z7

[0Z7]Prerequisites: [10X]↺↻ [0Z1]↺↻ [0Z3]↺↻.Difficulty:*.Let $K\subseteq {ℝ}^m$ compact. Consider the family of closed cubes with edge length $2^{-n}$ and centers at the grid points $2^{-n}{\mathbb{Z}}^m$. We call it ”n-tessellation”. Let $N_n$ be the number of cubes of the n-tessellation intersecting $K$. Show that $N_n$ is weakly increasing. Show that the following limit exists

if and only if the limit [(10.3)]↺↻ (that defines the dimension) exists. Show that, when they both exist, they coincide. This approach to computing the dimension is called Box Dimension.

These quantities have an interpretation in rate-distortion theory. ”$n$” is the position of the last significant digit (in base 2) in determining the position of a point $x$. ”${\log }_2{N}_n$” is the number of ”bits” needed to identify any $x\in K$ with such precision.