Algorithmic information, plane Kakeya sets, and conditional dimension

TL;DR

This paper introduces a new notion of conditional Kolmogorov complexity in Euclidean spaces, applying it to prove a lower bound on Hausdorff dimension for Kakeya sets and developing robust conditional dimensions with information-theoretic properties.

Contribution

It formulates conditional Kolmogorov complexity in Euclidean spaces and applies it to prove a Kakeya conjecture case and define robust conditional dimensions.

Findings

Proves a point-to-set principle linking point complexity to set dimension.

Provides a new proof of the two-dimensional Kakeya conjecture.

Develops robust lower and upper conditional dimensions with established relationships.

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 well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y).