Johan Jonasson, Chalmers.

The dynamical circle covering problem 

Abstract:

The classical circle covering problem is the following question:
Suppose that $I_1,I_2,I_3,\ldots$ are (closed) intervals of decreasing
lengths $l_1,l_2,\ldots$ and that these intervals are independently
and uniformly distibuted on the circle of unit circumference.  Then,
by Borel-Cantelli, any given point on the circle will a.s.  be covered
by infinitely many of these intervals iff the sum of the lengths is
divergent, but will the {\em whole} circle be covered?  By a famous
theorem of Schepp (1972) the answer is a.s. yes iff $\sum_{n=1}^\infty
\frac{e^{l_1+\ldots+l_n}}{n^2} = \infty$.  Assuming that $l_n = c/n$
for a constant $c$, this implies that the whole circle will be covered
infinitely often iff $c \geq 1$.

We will consider a dynamical version of the problem where the
intervals after having been given initial random positions move
according to independent standard Brownian motions.  Among other
things we will argue that for $c<2$ a.s. there are exceptional times
when a given fixed point is covered only finitely often, whereas this
is not the case when $c>2$.  We will also show that when $c<3$ there
are a.s. exceptional times where {\em some} point is covered only
finitely often, but when $c>3$ a.s the whole circle is covered all the
time.