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 curves 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, twicedifferentiable curves, graphs of smooth functions with m-bounded derivatives. In each case we get order-of-magnitude estimates; for example, there are twicedifferentiable 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 co-dimensions. Thus, twice-differentiable k-dimensional hypersurfaces in R may contain as many as OP (n ) 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. Several open mathematical questions are identified here for the attention of the probability community. Stylized applications in image processing and perceptual psychophysics are described.