Simply Normal Numbers to Different Bases

Let b ≥ 2 be an integer. A real number is called simply normal to base b if in its representation to base b every digit appears with the same asymptotic frequency. We answer the following question for arbitrary integers a, b ≥ 2:if a real number is simply normal to base a, does this imply that it is also simply normal to base b? It turns out that the answer is different from the well–known answers to the corresponding questions for the related properties "normality", "disjunctiveness", and "randomness". 1.) C. S. Calude, K. Salomaa, S. Yu (eds.). Advances and Trends in Automata and Formal Languages. A Collection of Papers in Honour of the 60th Birthday of Helmut Jürgensen.