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 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.
