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Sets of bounded discrepancy for multi-dimensional irrational rotation. (English) Zbl 1318.11097
Let $$\alpha=(\alpha_1, \ldots, \alpha_d)$$ be a vector in $$\mathbb{R}^d$$ and suppose that $$1, \alpha_1, \ldots, \alpha_d$$ are linearly independent over the rationals. Let $$\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$$. For a Riemann-measurable set $$S\subset \mathbb{T}^d$$ the discrepancy function of the sequence $$\{n\alpha\}$$ is defined by $$D_n(S,x)=\sum_{k=0}^{n-1}\chi_S(x+k\alpha)-n \operatorname{mes} S$$ for $$x\in \mathbb{T}^d$$, where $$\chi_S$$ is the indicator function of $$S$$. A measurable set $$S$$ is called a bounded remainder set (BRS) if there is a constant $$C=C(S,\alpha)$$ such that $$|D_n(S,x)|\leq C$$ for every $$n$$ and almost every $$x$$. The Hecke-Ostrowski-Kesten result states that in the case $$d=1$$ an interval $$I\subset \mathbb{T}$$ is a BRS if and only if its length belongs to $$\mathbb{Z}\alpha+\mathbb{Z}$$. The authors prove that any parallelepiped in $$\mathbb{R}^d$$ spanned by vectors belonging to $$\mathbb{Z}\alpha+\mathbb{Z}^d$$ is a BRS (Theorem 1). Two measurable sets $$S$$ and $$S'$$ in $$\mathbb{R}^d$$ are said to be equidecomposable if $$S$$ can be partitioned into finitely many measurable subsets that can be reassembled by rigid motions to form a partition of $$S'$$. The authors show that a Riemann measurable set $$S$$ in $$\mathbb{R}^d$$ is a BRS if and only if it is equidecomposable to some parallelepiped spanned by vectors in $$\mathbb{Z}\alpha+\mathbb{Z}^d$$, using translation by vectors belonging to $$\mathbb{Z}\alpha+\mathbb{Z}^d$$ (Corollary 3). Moreover in the case $$d=2$$ they give a characterization of the convex polygons with bounded remainder.

##### MSC:
 11K38 Irregularities of distribution, discrepancy 11J71 Distribution modulo one 52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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##### References:
 [1] U. Bolle. On multiple tiles in$$E$$\^{2}. Intuitive geometry. Proceedings of the 3rd international conference held in Szeged, Hungary, from 2 to 7 September 1991. North-Holland, Amsterdam; János Bolyai Mathematical Society, Budapest (1994), pp. 39-43. · Zbl 0818.52016 [2] Boltianski V.: Hilbert’s Third Problem. Wiley, New York (1978) [3] Cassels J.W.S.: An Introduction to the Geometry of Numbers. Springer, New York (1997) · Zbl 0866.11041 [4] C. De Concini, and C. Procesi. Topics in Hyperplane Arrangements, Polytopes and Box-Splines. Springer, New York (2011). · Zbl 1217.14001 [5] P. Erdős. Problems and results on diophantine approximations. Composito Mathematica, 16 (1964), 52-65 · Zbl 0131.04803 [6] Ferenczi, S., Bounded remainder sets, Acta Arithmetica, 4, 319-326, (1992) · Zbl 0774.11037 [7] Furstenberg, H.; Keynes, H.; Shapiro, L., Prime flows in topological dynamics, Israel Journal of Mathematics, 14, 26-38, (1973) · Zbl 0264.54030 [8] W.H. Gottschalk, and G.A. Hedlund. Topological dynamics. Colloquium Publications of the American Mathematical Society (AMS), Vol. 36. American Mathematical Society (AMS), Providence, Vol. VIII (1955), 151 p. · Zbl 0067.15204 [9] Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978) · Zbl 0408.14001 [10] H. Hadwiger. Translationsinvariante, additive und schwachstetige Polyederfunktionale. Archiv der Mathematik, 3 (1952), 387-394 (German) · Zbl 0048.28801 [11] H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Die Grundlehren der Mathematischen Wissenschaften, Vol. 93. Springer-Verlag, Berlin, Vol. XIII (1957), 312 S. · Zbl 0078.35703 [12] H. Hadwiger. Translative Zerlegungsgleichheit der Polyeder des gewöhnlichen Raumes. Journal fur die Reine und Angewandte Mathematik, 233 (1968), 200-212 (German) · Zbl 0167.19001 [13] H. Hadwiger, and P. Glur. Zerlegungsgleichheit ebener Polygone. Elemente der Mathematik, 6 (1951), 97-106 (German) [14] Halász, G., Remarks on the remainder in birkhoff’s ergodic theorem, Acta Mathematica Academiae Scientiarum Hungaricae, 28, 389-395, (1976) · Zbl 0336.28005 [15] S. Hartman. Colloquium Mathematicum, 1 (1947), 239-240 (French) · Zbl 0336.28005 [16] A. Haynes, and H. Koivusalo. Constructing Bounded Remainder Sets and Cut-and-Project Sets Which are Bounded Distance to Lattices (2014). arXiv:1402.2125. · Zbl 1341.11043 [17] E. Hecke. Über analytische Funktionen und die Verteilung von Zahlen mod. eins. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 1 (1921), 54-76 (German) · JFM 48.0184.02 [18] B. Jessen. Zur Algebra der Polytope. Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl., II (1972), 47-53 (German) · Zbl 0262.52004 [19] Jessen, B.; Thorup, A., The algebra of polytopes in affine spaces, Mathematica Scandinavica, 43, 211-240, (1978) · Zbl 0398.51009 [20] H. Kesten. On a conjecture of Erdős and Szüsz related to uniform distribution mod 1. Acta Arithmetica, 12 (1966), 193-212 · Zbl 0144.28902 [21] Liardet, P., Regularities of distribution, Composito Mathematica, 61, 267-293, (1987) · Zbl 0619.10053 [22] P. Mürner. Translative Zerlegungsgleichheit von Polytopen. Archiv der Mathematik, 29 (1977), 218-224 (German) · Zbl 0359.52007 [23] Oren, I., Admissible functions with multiple discontinuities, Israel Journal of Mathematics, 42, 353-360, (1982) · Zbl 0533.28009 [24] A. Ostrowski. Mathematische Miszellen IX: Notiz zur Theorie der Diophantischen Approximationen. Jahresber Dtsch Math-Ver, 36 (1927), 178-180 (German) · JFM 53.0165.02 [25] A. Ostrowski. Mathematische Miszellen. XVI: Zur Theorie der linearen diophantischen Approximationen. Jahresber Dtsch Math-Ver, 39 (1930), 34-46 (German) · JFM 56.0184.01 [26] Petersen, K., On a series of cosecants related to a problem in ergodic theory, Composito Mathematica, 26, 313-317, (1973) · Zbl 0269.10030 [27] G. Rauzy. Nombres algébriques et substitutions. Bulletin of the Society Mathematis of France, 110 (1982), 147-178 (French) · Zbl 0522.10032 [28] G. Rauzy. Ensembles à à restes bornés. Sémin. Théor. Nombres, Univ. Bordeaux I 1983-1984, Exp. No. 24. (1984), 12 p. (French). · Zbl 0533.28009 [29] C.-H. Sah. Hilbert’s third problem: scissors congruence. Research Notes in Mathematics, Vol. 33. San Francisco, London. Pitman Advanced Publishing Program, Vol. X (1979),188 p. · Zbl 0406.52004 [30] J. Schoissengeier. Regularity of distribution of ($$n$$α)-sequences. Acta Arithmetica, (2)133 (2008), 127-157 · Zbl 1228.11105 [31] Shephard, G.C., Combinatorial properties of associated zonotopes, Canadian Journal of Mathematics,, 26, 302-321, (1974) · Zbl 0287.52005 [32] A.V. Shutov. On a family of two-dimensional bounded remainder sets. Chebyshevskiĭ Sb., (4 (40))12 (2011), 264-271 (Russian) · Zbl 1302.11048 [33] J.-P. Sydler. Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à à trois dimensions. Commentarii Mathematici Helvetici, 40 (1965), 43-80 (French) · Zbl 0135.20906 [34] P. Szüsz. Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats. Acta Mathematica Academiae Scientiarum Hungaricae, 5 (1954), 35-39 (German) · Zbl 0058.03503 [35] P. Szüsz. Lösung eines Problems von Herrn Hartman. Studia Mathematica, 15 (1955), 43-55 (German) · Zbl 0067.02401 [36] Zhuravlev, V.G., Bounded remainder polyhedra, Proceedings of the Steklov Institute of Mathematics, 280, 71-90, (2013) · Zbl 1370.11092 [37] Ziegler G.M.: Lectures on Polytopes. Springer-Verlag, Berlin (1995) · Zbl 0823.52002
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