On a problem of Schweiger concerning normal numbers (Q5929409)

Let \(T\) and \(S\) be two number theoretical transformations on the unit \(n\)-cube, which are defined by \textit{F. Schweiger} [J. Number Theory 1, 390-397 (1969; Zbl 0184.07503)]. Write \(T\sim S\) if there exist positive integers \(m\) and \(n\) such that \(T^m=S^n\). Schweiger's main result states that \(T\sim S\) implies that every \(T\)-normal number \(x\) is \(S\)-normal. Furthermore, he conjectured that \(T\not\sim S\) implies that not all \(T\)-normal \(x\) are \(S\)-normal. Two counter-examples to this conjecture are given in this paper. In two cases of binary expansion modified and of golden mean expansions modified, \(T\not\sim S\) and an irrational number \(x\) is \(T\)-normal if and only if \(x\) is \(S\)-normal.