2025/10/13 by Fiedler, Jacob B.
#28A78 #28A80 #68Q30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2510.11645
We prove a new lower bound on the algorithmic information content of points lying on a line in ℝn. More precisely, we show that a typical point z on any line ℓ satisfies Kr(z)≥ (Kr(ℓ))/(2) + r - o(r) 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.