On the construction of absolutely normal numbers
Volume 180 / 2017
Acta Arithmetica 180 (2017), 333-346
MSC: Primary 11K16; Secondary 11Y16, 68-04.
DOI: 10.4064/aa170213-5-8
Published online: 28 September 2017
Abstract
We give a construction of an absolutely normal real number $x$ such that for every integer $b \ge 2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order $\mathcal{O}(N^{-1/2})$. 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.
