On the discrepancy of sequences associated with the sum-of-digits function (Q1080885)

Let \(w:=(q_ n)_{n\in {\mathbb{N}}}\) be the sequence of best approximation denominators of a real number \(\alpha\). Every integer N has an unique digit representation with respect to this base w. If \(s_{\alpha}(N)\) denotes the digit-sum of N with respect to w, then for every irrational x, the sequence \((s_{\alpha}(n)\cdot x)_{n\in {\mathbb{N}}}\) is uniformly distributed modulo one. Estimates for the discrepancy of this sequence are given which turn out to be best possible in the case that \(\alpha\) has bounded continued fraction coefficients.