Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

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 ).