On absolutely normal numbers and their discrepancy estimate

Abstract

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^{0}x,b^{1}x,b^{2}x,\dots )$ is essentially the same as that realized by almost all real numbers.

This paper has been superseded by: "On the construction of absolutely normal numbers" by Christoph Aistleitner, Verónica Becher, Adrian-Maria Scheerer and Theodore Slaman, July 2017, arXiv:1707.02628, http://arxiv.org/abs/1707.02628