Mutual Dimension

Abstract

We define the lower and upper mutual dimensions mdim(x : y) and Mdim(x : y) between any two points x and y in Euclidean space. Intuitively, these are the lower and upper densities of the algorithmic information shared by x and y. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that if f : R m → R n is computable and Lipschitz, then the inequalities mdim( f (x) : y) ≤ mdim(x : y) and Mdim( f (x) : y) ≤ Mdim(x : y) hold for all x ∈ R m and y ∈ R t . We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.

Bounding the Dimension of Points on a Line

“…where dim H (E) is the (classical) Hausdorff dimension of E, i.e., the most standard notion of fractal dimension. Although (1) does not hold in general, this correspondence suggested that effective dimensions might provide new techniques for dimension bounds in classical fractal geometry. A recent point-to-set principle of J. Lutz and N. Lutz [14] reinforces that prospect by characterizing the Hausdorff dimension of arbitrary sets in terms of effective dimension.…”

Mutual Dimension and Random Sequences

“…The theorem follows directly from the properties of mutual dimension between points in Euclidean space found in [4] and the correspondences described in corollaries 2.5, 2.6, and 2.14.…”

“…The objects x and y of interest in [4] are points in Euclidean spaces R n and their images under computable functions, so the fine-scale geometry of R n plays a major role there.…”

“…We thank an anonymous reviewer of [4] for posing the question answered by Corollary 3.15. We also thank anonymous reviewers of this paper for useful comments, especially including Observation 3.2.…”

Dimension Spectra of Lines

“…Second, the dimension spectrum of a line L a,b ⊆ R 2 may properly contain the unit interval described in our main theorem, even when dim(a, b) = Dim(a, b). If a ∈ R is random and b = 0, for example, then sp(L a,b ) = {0} ∪ [1,2]. It is less clear whether this set of "exceptional values" in sp(L a,b ) might itself contain an interval, or even be infinite.…”