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BR-sets. (Russian. English summary) Zbl 1436.11095
Let $$\alpha_1,\ldots,\alpha_d$$ be irrational numbers and suppose that $$1,\alpha_1,\ldots,\alpha_d$$ are linearly independent over $$\mathbb{Z}$$. The set $$X\subset [0;1)^d$$ is called a bounded remainder set if the quantity $\delta(i,x_0)=\sharp\{ j:0\leq j<i:(\{j\alpha_1+x_0\},\ldots,\{j\alpha_d+x_0\})\in X\}-ivol(X)$ is bounded by an absolute constant.
The paper is an expository of previous author’s results on bounded remainder sets for $$d=2$$ and $$d=3$$, mainly devoted to constructing new examples of bounded remainder sets and finding exact bounds for $$\delta(i,x_0)$$ for these sets.

##### MSC:
 11K38 Irregularities of distribution, discrepancy 11J71 Distribution modulo one
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##### References:
 [1] Weyl H., “Über die Gibbs”sche Erscheinung und verwandte Konvergenzph änomene”, Rendicontidel Circolo Mathematico di Palermo, 30 (1910), 377-407 · JFM 41.0528.02 [2] Hecke E., “Eber Analytische Funktionen und die Verteilung von Zahlen mod. eins”, Math. Sem. Hamburg. Univ., 5 (1921), 54-76 · JFM 48.0197.03 [3] Erdös P., “Problems and results on diophantine approximation”, Comp. Math., 16 (1964), 52-65 · Zbl 0131.04803 [4] Kesten H., “On a conjecture of Erdös and Szüsz related to uniform distribution mod 1”, Acta Arithmetica, 12 (1966), 193-212 · Zbl 0144.28902 [5] Zhuravlev V. G., “One-dimensional Fibonacci tilings”, Izvestiya: Mathematics, 71:2 (2007), 89-122 · Zbl 1168.11006 [6] Shutov A. V., “Optimum estimates in the problem of the distribution of fractional parts of the sequence $$n\alpha$$”, Vestnik SamGU. Yestestvennonauchnaya seriya, 5:3 (2007), 112-121 [7] Krasilshchikov V. V., Shutov A. V., “Description and exact maximum and minimum values of the remainder in the problem of the distribution of fractional parts”, Mathematical Notes, 89:1 (2011), 43-52 · Zbl 1303.11078 [8] Oren I., “Admissible functions with multiplie discontinioutes”, Israel J. Math., 42 (1982), 353-360 · Zbl 0533.28009 [9] Shutov A. V., “The two-dimensional Hecke-Kesten problem”, Chebyshevskiy sbornik, 12:2 (2011), 151-162 · Zbl 1306.11055 [10] Szüsz R., “Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats”, Acta Math. Acad. Sci. Hungar., 1954, no. 5, 35-39 · Zbl 0058.03503 [11] Liardet P., “Regularities of distribution”, Compositio Math., 61 (1987), 267-293 · Zbl 0619.10053 [12] Rauzy G., “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147-178 · Zbl 0522.10032 [13] Ferenczi S., “Bounded remainder sets”, Acta Arithmetica, 61 (1992), 319-326 · Zbl 0774.11037 [14] Zhuravlev V. G., “Rauzy tilings and bounded remainder sets on the torus”, Journal of Mathematical Sciences, 322 (2005), 83-106 · Zbl 1158.11331 [15] Zhuravlev V. G., “Exchanged toric developments and bounded remainder sets”, Journal of Mathematical Sciences, 392 (2011), 95-145 [16] Zhuravlev V. G., “Multidimensional Hecke theorem on the distribution of fractional parts”, St. Petersburg Mathematical Journal, 24:1 (2012), 1-33 [17] Zhuravlev V. G., “Bounded remainder polyhedra”, Proc. Steklov Inst. Math., 280:suppl. 2 (2013), S71-S90 · Zbl 1370.11092 [18] Abrosimova A. A., “Bounded remainder sets on a two-dimensional torus”, Chebyshevskiy sbornik, 12:4(40) (2011), 15-23 [19] Abrosimova A. A., “Average values for deviation distribution of points on the torus”, Belgorod State University Scientific bulletin. Mathematics & Physics, 5(124):26 (2012), 5-11 [20] Abrosimova A. A., “Boundaries of deviations for three-dimensional bounded remainder sets”, Belgorod State University Scientific bulletin. Mathematics & Physics, 19(162):32 (2013), 5-21 [21] Abrosimova A. A., “Fractal bounded remainder sets”, Mathematical Modeling of Fractal Processes of Analysis and Informatics, II Int. Conf. Proc. of Young Scientists (Nalchik, 2012), 18-21 [22] Zhuravlev V. G., “Geometrization of Hecke”s theorem”, Chebyshevskiy sbornik, 11:1 (2010), 125-144 · Zbl 1345.11058 [23] Abrosimova A. A., Blinov D. A., Polyakova T. V., “Optimization of boundaries of remainder for bounded remaider sets on two-dimensional torus”, Chebyshevskiy sbornik, 14:1(45) (2013), 9-17 · Zbl 1270.41006 [24] Abrosimova A. A., Blinov D. A., “Boundaries optimization of two-dimensional bounded remainder sets”, Belgorod State University Scientific bulletin. Mathematics & Physics, 26(169):33 (2013), 5-13 [25] Abrosimova A. A., Zhuravlev V. G., “Hecke theorem generalization to the two-dimensional case and balanced words”, Algebra and Number Theory: modern problems and applications, VIII Int. Conf. Proc., dedicated to 190-year anniversary of the P. L. Chebyshev and 120-year anniversary of the I. M. Vinogradov (Saratov, 2011), 3-4 [26] Abrosimova A. A., “Multiplication of toric developments and constructing of bounded remainder sets”, Uchenye zapiski Orlovskogo Gosudarstvennogo Universiteta. Seriya: Yestestvennyye, tekhnicheskiye i meditsinskiye nauki, 6:2 (2012), 30-37
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