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Connect the dots: how many random points can a regular curve pass through?

2002
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Advances in Applied Probability
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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 O P (n 1/3) uniform random points, but not essentially more than this. More generally,

doi:10.1239/aap/1127483737
fatcat:ns3ek77uqfhyrfvugfdmam5jyu