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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)].

11K38 Irregularities of distribution, discrepancy
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