On a question of Mendès France on normal numbers

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 O(n4) mathematical operations.