The information content of points on lines and $k$-plane extensions

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-Cs\"ornyei-Lutz-Lutz-Mayordomo-Stull.