A problem in multi-index normality (Q1378977)

The normality of the number \(x\) to the base \(r\) is equivalent to the uniform distribution (u.d.) mod 1 of the sequence \((r^nx: n\in \mathbb{N})\). The authors establish a general `multi-index' normality theorem, which states that for \(r>2\) and \(K\) non-zero integers \(s_1,\dots, s_K\) with each \(s_i/s_j\) a power of \(r,x\) is normal to base \(r\) if and only if the following extended notion of u.d. mod 1 holds: for all subintervals \(I\) of \([0,1]\) \[ \frac{1}{N^K}\Biggl| (n_j)\in \mathbb{N}^K: n_j\leq N,\;\Biggl\{ \sum_{j=1}^K s_jr^{n_j}+ r^{m_j}x\Biggr\}\in I\Biggr|\to| I| \] as \(N\to\infty\) (\(\{x\}\) denotes the usual fractional part of \(x\) and \(| . |\) the appropriate measure of the sets). If any of the ratios \(s_i/s_j\) is not a power of \(r\), then there are uncountably many \(x\) not normal in base \(r\) but u.d. mod 1 in the above extended sense. It is readily seen that the normality of \(x\) to the base ring \(r\) implies that \(((sr^n+r^m)x: m,n\in \mathbb{N})\) is u.d. mod 1 in the above sense and it was asked whether the converse held. It is a simple consequence of the above theorem that it is not always true. As well as classical methods (Weyl's criterion, van der Corput's difference theorem), the authors use Riesz measures in techniques that they, with others, have exploited to good effect in this area.