zbMATH — the first resource for mathematics

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.

11K38 Irregularities of distribution, discrepancy
11J71 Distribution modulo one
Full Text: DOI MNR
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.