The information content of points on lines and $k$-plane extensions
Jacob B. Fiedler
TL;DR
The paper establishes a new lower bound on the information content of a typical point on a line in $\mathbb{R}^n$, showing $K^A_r(\ell(s)) \ge \tfrac{1}{2}K^A_r(\ell) + r - o(r)$ under a randomness assumption on $s$ relative to an oracle $A$. The key technique combines a surrogate-point construction with the point-to-set principle to translate line-level complexity into bounds on Hausdorff dimension growth for unions of $k$-planes. This yields a classical bound: if $E \subseteq \mathbb{R}^n$ and $F$ is the union of $E$ with every $k$-plane that intersects $E$ in positive measure, then either $E=F$ or $\dim_H(F) \le 2\dim_H(E) - k$. The results connect algorithmic information theory to geometric measure theory, refining surrogate-point methods and contributing to the understanding of dimension growth under $k$-plane extensions, with implications for Furstenberg-type sets and related projection problems.
Abstract
We prove a new lower bound on the algorithmic information content of points lying on a line in $\mathbb{R}^n$. More precisely, we show that a typical point $z$ on any line $\ell$ satisfies \begin{equation*} K_r(z)\geq \frac{K_r(\ell)}{2} + r - o(r) \end{equation*} at every precision $r$. In other words, a randomly chosen point on a line has (at least) half of the complexity of the line plus the complexity of its first coordinate. We apply this effective result to establish a classical bound on how much the Hausdorff dimension of a union of positive measure subsets of $k$-planes can increase when each subset is replaced with the entire $k$-plane. To prove the complexity bound, we modify a recent idea of Cholak-Csörnyei-Lutz-Lutz-Mayordomo-Stull.