EDB — 0Z7

[0Z7]Prerequisites: [10X]↺↻ [0Z1]↺↻ [0Z3]↺↻.Difficulty:*.Let \(K⊆ ℝ^ m\) compact. Consider the family of closed cubes with edge length \(2^{-n}\) and centers at the grid points \(2^{-n}ℤ^ m\). 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 \(x∈ K\) with such precision.