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Bounded remainder sets. (English) Zbl 0774.11037
The author proves a necessary and sufficient condition for bounded remainder sets. The technique is based on an approach developed by G. Rauzy [Sémin. Théor. Nombres, Univ. Bordx. 1983/84, Exp. No. 24 (1984; Zbl 0547.10044)]. Let $$L$$ be a lattice in $$\mathbb{R}^ s$$ and let $$\alpha$$ be an element of $$\mathbb{R}^ s$$. Define $$X:=\mathbb{R}/L$$ and $$T:X\to X$$, $$Tx:=x+\alpha\bmod L$$. $$T$$ preserves Lebesgue measure $$\lambda$$ on $$X$$. A subset $$A$$ of $$\mathbb{R}^ s$$ is called $$L$$-simple if the condition $$x,y\in A$$, $$x-y\in L$$ implies $$x=y$$. A subset $$A$$ of $$X$$ is called a bounded remainder set (BRS), if there exist real numbers $$a$$ and $$C$$ such that, for all positive integers $$n$$, $\left| \sum_{k=0}^{n-1} \mathbf{1}_ A(T^ k x)-na\right|\leq C.$ If we identify $$L$$- simple subsets $$A$$ of $$\mathbb{R}^ s$$ with their projections onto $$X$$, then the definition of a BRS applies to these sets as well. It is well-known that, for measurable sets $$A$$, $$A$$ is a BRS if and only if the function $$\mathbf{1}_ A(\cdot)-a$$ is a coboundary [for this “coboundary theorem” in its general form and further helpful references, see P. Liardet, Compos. Math. 61, 267-293 (1987; Zbl 0619.10053)]. The result of this paper, expressed in the second of two forms given by the author, is the following. Let $$X$$ and $$T$$ be as above. A measurable subset $$A$$ of $$\mathbb{R}^ s$$ with nonempty interior is a BRS if and only if there is a lattice $$M$$ in $$\mathbb{R}^ s$$, an element $$\beta\in\mathbb{R}^ s$$ and a partition of $$A$$ into finitely many measurable sets $$B_ i$$ of positive measure such that (i) the couple $$(\beta,M)$$ is minimal, (ii) $$Sx- x\in\mathbb{Z}\beta+M$$ for almost all $$x$$, and (iii) $$S_ ix\equiv x+k\beta\bmod M$$ whenever $$S_ i=S^ k$$. Here $$S$$ denotes the induced transformation on $$A$$ and $$S_ i$$ the induced transformation on $$B_ i$$.
The list of known results about BRS given in this article should be completed by references to the papers of K. Petersen [Compos. Math. 26, 313-317 (1973; Zbl 0269.10030)], L. Shapiro [Studies in probability and ergodic theory, Adv. Math., Suppl. Stud., Vol. 2, 135-154 (1978; Zbl 0446.10045)] and G. Halasz [Acta Math. Acad. Sci. Hung. 27, 389-395 (1976; Zbl 0336.28005)].

##### MSC:
 11K38 Irregularities of distribution, discrepancy
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