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Regularities of distribution. (English) Zbl 0619.10053
A subset A of a compact metrizable space X is said to be a bounded remainder set (b.r.s.) for a sequence \((x_ n)\), \(x_ n\in X\) if there exists a, \(0\leq a<1\), such that \(card\{m<n;\quad y_ m\in A\}-na,\) \(n\in {\mathbb{N}}\) is bounded. The case \(X={\mathbb{R}}/{\mathbb{Z}}\), \(x_ n=n\alpha\) (mod 1), A an interval (mod 1), was first considered by Hecke and completely solved by H. Kesten [Acta Arith. 12, 193-212 (1966; Zbl 0144.289)]. Generalizations were given by H. Furstenberg, H. Keynes, L. Shapiro, R. Rourrence and K. Petersen [Compos. Math. 26, 313-317 (1973; Zbl 0269.10030)] and I. Oren [Isr. J. Math. 42, 353-360 (1982; Zbl 0533.28009)].
This paper starts with a general Coboundary theorem and an improvement for linear isometries on a Hilbert space (Theorem 1). Theorem 2 can be viewed as the metric version of a classical theorem of W. H. Gottschalk and G. A. Hedlund [Topological dynamics (1955; Zbl 0067.152)]. Theorem 3 gives an analogon of Kesten’s result, for m-dimensional intervals in \(({\mathbb{R}}/{\mathbb{Z}})^ m\). Theorem 4 gives some results for more general subsets of \(({\mathbb{R}}/{\mathbb{Z}})^ m\) (extending results of Szüsz and Rauzy, the proof follows an idea of Larcher). Theorem 5 shows that there exist no nontrivial intervals which are b.r.s. of Weyl sequences \(x_ n=p(n)\), \(p(x)=a_ nx^ n+...+a_ 0\), \(n\geq 2\), \(a_ n\) irrational.
The final section deals with q-multiplicative sequences using results of Coquet, Kamae, Mendès-France, extending results of M. Queffelec [Contribution à l’étude spectrale de suites arithmétiques (Thèse d’Etat, Université Paris-Nord) (1984)] (Theorem 6). The related measured flows are described (Theorem 7, 8). Some applications conclude this interesting paper.
Reviewer: H.Rindler

MSC:
11K38 Irregularities of distribution, discrepancy
11K06 General theory of distribution modulo \(1\)
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28D99 Measure-theoretic ergodic theory
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