On the construction of absolutely normal numbers

We give a construction of an absolutely normal real number $x$ such that for every integer $b$ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^{n}x\mathrm{m}\mathrm{o}\mathrm{d}1{)}_{n\ge 0}$ is of asymptotic order $\mathcal{O}(N^{-1\mathrm{/}2})$. 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 was not known before.