Fractals Generated by Algorithmically Random Brownian Motion

A continuous function x on the unit interval is an algorithmically random Brownian motion when every probabilistic event A which holds almost surely with respect to the Wiener measure, is reflected in x, provided A has a suitably effective description. In this paper we study the zero sets and global maxima from the left as well as the images of compact sets of reals of Hausdorff dimension zero under such a Brownian motion. In this way we shall be able to find arithmetical definitions of perfect sets of reals whose elements are linearly independent over the field of recursive real numbers.