On rings of numbers which are normal to one base but non-normal to another (Q1900878)

The existence of real numbers that are normal to a given integer base \(g\geq 2\) but are non-normal to another integer base \(h\geq 2\) when \(g\), \(h\) are multiplicatively independent has been known for some time. The author observed that the number given by the series \(\sum_{n=1}^\infty 2^{-n} 5^{-4^n}\), shown to be normal to base 5 by \textit{R. G. Stoneham} [Acta Arith. 22, 371-389 (1973; Zbl 0276.10029)], is in fact non-normal to base 10. This observation was the basis of the striking result that there is an uncountable subring \(R\) of \(\mathbb{R}\) in which the non-zero elements are normal to base \(g\) but all the elements are non- normal to the base \(pg\), where \(p\) is an odd prime not dividing \(g\). This implies that given a nonconstant \(q\in \mathbb{Z} [x]\) there are explicitly given numbers \(x\) such that \(q(x)\) are transcendental, normal to base 3 and non-normal to base 15. In particular, this holds for \(x^2, x^3, \dots\;\). Some related questions and extensions to more general pairs of bases \((g, h)\) are discussed but the method requires the divisibility condition \(g\mid h\). Following the untimely death of the author Gerold Wagner, this paper was revised by Bodo Volkmann with the assistance of Soon Man-Jung, who have done the mathematical community a considerable service by ensuring that these remarkable results have been published.