On a question of Mend\`es France on normal numbers

TL;DR

This paper constructs a computable real number x such that both x and 1/x are normal in all integer bases and in their continued fraction expansions, answering Mendès France’s question from 2008.

Contribution

It provides the first explicit construction of a number normal in all bases and continued fractions, with both x and 1/x being computable.

Findings

x and 1/x are both continued fraction normal

x and 1/x are normal in all integer bases ≥ 2

both numbers are computable

Abstract

In 2008 or earlier, Michel Mend\`es 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 both computable.