Connect-The-Dots: How many random points can a regular curve pass through?

Suppose n points are scattered uniformly at random in the unit square [0, 1]. Question: How many of these points can possibly lie on some curve of length λ? Answer, proved here: OP (λ · √ n). We consider a general class of such questions; in each case, we are given a class Γ of curves in the square, and we ask: in a cloud of n uniform random points, how many can lie on some curve γ ∈ Γ? Classes of interest include (in addition to the rectifiable curves mentioned above): Lipschitz graphs, monotone graphs, twice-differentiable curves, graphs of smooth functions with m-bounded derivatives. In each case we get order-of-magnitude estimates; for example, there are twice-differentiable curves containing as many as OP (n) uniform random points, but not essentially more than this. We also consider generalizations to higher dimensions and to hypersurfaces of various codimensions. Thus, twice-differentiable k-dimensional hypersurfaces in R may contain as many as OP (nk/(2d−k)) uniform random points. We also consider other notions of ‘passing through’ such as passing through given space/direction pairs. Thus, twice-differentiable curves in R may pass through at most OP (n) uniform random location/direction pairs. We give both concrete approaches to our results, based on geometric multiscale analysis, and abstract approaches, based on ε-entropy. Stylized applications in image processing and perceptual psychophysics are described. AMS subject classifications. Primary 60D05; secondary 62M40, 54C70, 62G10, 60G55.