Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

Abstract

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways.1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. 2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the wellstudied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x : y) and Mdim(x : y).

Who Asked Us? How the Theory of Computing Answers Questions about Analysis

“…J. Lutz and N. Lutz gave computability theoretic proofs of both of these facts. They showed that the former corresponds to the existence of lines in all directions that contain no random points [25], and that the latter corresponds to the fact that for any random pair (a, x) ∈ R 2 , dim(x, ax + b) = 2 holds for all b ∈ R [26].…”

Computability and Randomness

“…The most recent and powerful of these is the following, where we denote the classical Haus-dorff dimension of ⊆ ℝ by dim H ( ), and its classical packing dimension by dim p ( ). [19]).…”

“…For example, Lutz and Stull [21] obtained a new lower bound on the Hausdorff dimension of generalized sets of Furstenberg type; Lutz [20] showed that a fundamental intersection formula, due in the Borel case to Kahane and Mattila, is true for arbitrary sets; and Lutz and Lutz [19] gave a new proof of the two-dimensional case (originally proved by Davies) of the well-known Kakeya conjecture, which states that, for all ≥ 2, if a subset of ℝ has lines of length 1 in all directions, then it has Hausdorff dimension .…”

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

“…In computability theory (particularly in algorithmic randomness theory), it is usual to consider the algorithmic dimension (the algorithmic information density) of a point as an analogue of fractal dimension; see [7,Section 13]. The notions of Kolmogorov complexity and algorithmic dimension in a Euclidean space has been studied in [22][23][24][25], for instance. In this section, we compare our notion of topological dimension of points and the notions of effective fractal dimension of points.…”