A critical case for Brownian slow points

Let X(t) be a Brownian motion and let S(c) be the set of reals r greater than or equal to 0 such that \X(r+l)-X(r)\ less than or equal to c root t, 0 less than or equal to t less than or equal to h, for some h = h(r) > 0. It is known that S(c) is empty if c < 1 and nonempty if c > 1, a.s. In this paper we prove that S(1) is empty a.s.