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Connect the Dots: How Many Random Points Can a Regular Curve Pass Through?

Ery Arias-Castro, David L. Donoho, Xiaoming Huo and Craig A. Tovey
Advances in Applied Probability
Vol. 37, No. 3 (Sep., 2005), pp. 571-603
Stable URL: http://www.jstor.org/stable/30037345
Page Count: 33
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Connect the Dots: How Many Random Points Can a Regular Curve Pass Through?
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Abstract

Given a class Γ of curves in $[0, 1]^2$, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as Op($n^{1/3}$) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes $[0, 1]^d$ and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in $R^d$ may contain as many as Op ($n^{k/2d-k)}$) uniform random points. We also consider other notions of 'incidence', such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in $R^2$ may pass through at most Op($n^{1/4}$) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

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