Dynamical circel covering with homogeneous Poisson updating

Journal article, 2008

We consider a dynamical variant of Dvoretzky's classical covering problem of the unit circumference circle, where the centers of the arcs are updated according to independent Poisson processes of unit intensity. This dynamical model was introduced (in greater generality) in Jonasson and Steif [Jonasson, J., Steif, J., 2008. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36, 739-764], where is was shown that when the length of the n'th arc is ℓn and we write un {colon equals} ∏k = 1n (1 - ℓk), then lim infn n (log n) un < ∞ implies that the whole circle is a.s. covered at all times, whereas if ∑n eℓ1 + ⋯ + ℓn / (n2 log n) < ∞, then a.s. there are times at which the circle is not fully covered. In this paper we modify the former condition to lim supn n un < ∞. In particular this takes care of the natural border case ℓn = 1 / n; no exceptional times exist. More generally, with the parametrization ℓn = c / n, there is no c for which there are exceptional times for which the model behaves differently than for the static case.