Abstract

We prove that the Hausdorff dimension of the set of normal numbers in base 2 whose binary expansions have the property that for any k, the k-th digits and the 2k-th digits are not all equal to 1, is one-half. Further, we obtain the Hausdorff dimension formula for the set of normal numbers in base 2 with given limit of multiple ergodic averages of some special function. This is a joint work with Michal Rams.