We prove a new lower bound on the algorithmic information content of points lying on a line in Euclidean space. Roughly speaking, we show that 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-Csornyei-Lutz-Lutz-Mayordomo-Stull.
