On the construction of absolutely normal numbers

We give a construction of an absolutely normal real number x such that for every integer b≥2, the discrepancy of the first N terms of the sequence (bnxmod1)n≥0 is of asymptotic order O(N−1/2). This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order has not been known before.