Regularities of the distribution of \(\beta\)-adic van der Corput sequences
Let \(x_n\), \(n=0,1,\dots\), be a sequence in \([0,1)\), \(I\) be an interval in \([0,1]\), \[ D(N,I)=| \#\{0\leq n<N-1;x_n\in I\}-N| I| | \] and \(| I| \) denote the length of \(I\). For \(D(N,I)\), \(N\to\infty\), we have: (i) For every sequence \(x_n\) there exists interval \(I\subset[0,1]\) for which \(D(N,I)\) is unbounded (T. Van Aardenne-Ehrenfest, 1949); (ii) The set of intervals \(I\) for which \(D(N,I)\) is bounded is at most countable (W.M. Schmidt, 1974); (iii) For \(x_n=n\alpha\bmod1\), \(\alpha\) irrational, the \(D(N,I)\) is bounded if and only if \(I=k\alpha\bmod1\) for some integer \(k\) (E. Hecke, 1921); (iv) If \(x_n\) is van der Corput sequence in base \(q\), then \(D(N,I)\) is bounded if and only if \(| I| \) has finite \(q\)-ary expansion (W.M. Schmidt, 1974); (v) If \(x_n=\alpha s_q(n)\), \(\alpha\) irrational and \(s_q(n)\) is the sum-of-digit function, then the only intervals \(I\) with bounded \(D(N,I)\) are the trivial ones \(| I| =0\) or \(| I| =1\) (P. Liardet, 1987). Exact references can be found in [\textit{O. Strauch} and \textit{Š. Porubský}, Distribution of Sequences: A Sampler. Schriftenreihe der Slowakischen Akademie der Wissenschaften 1. Bern: Peter Lang. (2005; Zbl 1078.11051)]. In this paper the author determines intervals \(I\) with bounded \(D(N,I)\) for \(\beta\)-adic van der Corput sequences. He proves: (vi) If \(\beta\) is a Parry number and \(D(N,I)\) is bounded, then \(| I| \in\mathbb Q(\beta)\); (vii) If \(\beta\) is a Pisot (special) number, then \(D(N,[0,y))\) is bounded if and only if the \(\beta\)-expansion of \(y\) is finite or its tail is the same as that of \(\beta\)-expansion of \(1\). Here: \(\beta\) is a Parry number if the expansion of \(1\) is finite or eventually periodic; \(\beta\)-adic van der Corput sequence is a sequence of \(x_n\in[0,1)\) with finite \(\beta\)-expansion ordered ``inverse'' lexicographically (see [\textit{S. Ninomiya}, RIMS Kokyuroku 1032, 129--141 (1998; Zbl 1044.11599)]); and basic properties of \(\beta\)-expansion can be found in \textit{W. Parry} [Acta Math. Acad. Sci. Hung. 11, 401--416 (1960; Zbl 0099.28103)].
scientific article; zbMATH DE number 5126991
scientific article