Bounding the Dimension of Points on a Line

TL;DR

This paper employs Kolmogorov complexity to establish lower bounds on the effective Hausdorff dimension of points on a line and applies these results to improve bounds in fractal geometry, specifically for Furstenberg-type sets.

Contribution

It introduces a novel method using Kolmogorov complexity to derive lower bounds on the dimension of points on a line, with applications to fractal geometry.

Findings

Lower bounds on effective Hausdorff dimension for (x, ax+b)

Improved lower bounds on Hausdorff dimension of Furstenberg sets

Application of complexity methods to geometric measure theory

Abstract

We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point (x, ax+b), given real numbers a, b, and x. We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.