On a question of Mendès France on normal numbers
Volume 203 / 2022
Acta Arithmetica 203 (2022), 271-288
MSC: Primary 11K16, 11J70.
DOI: 10.4064/aa210813-28-1
Published online: 15 April 2022
Abstract
In 2008 or earlier, Michel Mendès France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such that both $x$ and its reciprocal $1/x$ are continued fraction normal as well as normal to all integer bases greater than or equal to $2$. Moreover, $x$ and $1/x$ are computable, the first $n$ digits of their continued fraction expansion can be obtained in $\mathcal {O}(n^4)$ mathematical operations.
