Fractal Intersections and Products via Algorithmic Dimension

TL;DR

This paper uses algorithmic fractal dimensions based on Kolmogorov complexity to establish bounds on Hausdorff and packing dimensions of intersections and products of fractals, extending classical results to arbitrary sets.

Contribution

It introduces a novel approach linking algorithmic dimensions to classical fractal dimensions, broadening the scope of dimension bounds to all sets.

Findings

Bounds on Hausdorff and packing dimensions derived from algorithmic dimensions

Classical dimension results extended to arbitrary sets

Method bridges algorithmic information theory and fractal geometry

Abstract

Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.