On the distribution function of the remainder term on bounded remainder sets
Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 90-107.

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Bounded remainder sets are sets with bounded by constant independent of the number of points remainder term of the multidimensional problem of the distribution of linear function fractional parts. These sets were introduced by Hecke and studied by Erdös, Kesten, Furstenberg, Petersen, Szusz, Liardet and others. Currently, in one-dimensional case full description of bounded remainder intervals and exact estimates of the remainder term on such intervals are known. Also some more precise results about the remainder term are established. Among these results there are exact formulaes for maximum, minimum and average value of the remainder term, description of the remainder term as piecewise linear function, non-monotonic estimates for the remainder term, estimates of speed of attainment of the remainder term exact boundaries, etc …In the higher dimensional cases only several examples of bounded remainder sets are known. Particularly, in recent years V. G. Zhuravlev, A. V. Shutov, and A. A. Abrosimova introduce a new construction of some families of multidimensional bounded remainder sets based on exchanged toric tilings. For introduced sets we are able not only to prove the boundness of the remainder term but to compute exact values of its minimum, maximum, and average. In the present work we study more subtle property of the remainder term on bounded remainder sets based on exchanged toric tilings: its distribution function. It is proved that the remainder term is uniformly distributed only in one-dimensional case. An algorithm for computation of the normalized distribution function is given. Some structural results about this function are proved. For some two-dimensional families of bounded remainder sets their normalized distribution functions are clealy calculated. Bibliography: 31 titles.
Keywords: distribution modulo one, bounded remainder sets, exchanged toric tilings, distribution function.
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A. A. Zhukova; A. V. Shutov. On the distribution function of the remainder term on bounded remainder sets. Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 90-107. http://geodesic.mathdoc.fr/item/CHEB_2016_17_1_a6/

[1] Bohl P., “Über ein in der Theorie der säkutaren Störungen vorkommendes Problem”, J. Reine Angew. Math., 135 (1909), 189–283 | MR | Zbl

[2] Day Bradley A., “Prismatoid, Prismoid, Generalized Prismoid”, The American Math. Monthly, 86 (1979), 486–490 | MR

[3] Erdös P., “Problems and results on diophantine approximations”, Compositio Math., 16 (1964), 52–65 | MR | Zbl

[4] Furstenberg H., Keynes M., Shapiro L., “Prime flows in topological dynamics”, Israel J. Math., 14 (1973), 26–38 | DOI | MR | Zbl

[5] Grepstad S., Lev N., “Sets of bounded discrepancy for multi-dimensional irrational rotation”, Geometric and Functional Analysis, 25:1 (2015), 87–133 | DOI | MR | Zbl

[6] Hecke E., “Eber Analytische Funktionen und die Verteilung van Zahlen mod Eins”, Math. Sem. Hamburg Univ., 5 (1921), 54–76

[7] Heynes A., Koivusalo H., “Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices”, Israel J. Math. (to appear) , arXiv: 1402.2125

[8] Kelly M., Sadun L., “Patterns equivariant cohomology and theorems of Kesten and Oren”, Bull. London Math. Soc., 47:1 (2015), 13–20 | DOI | MR | Zbl

[9] Kesten H., “On a conjecture of Erdös and Szüsz related to uniform distribution mod 1”, Acta Arithmetica, 12 (1966), 193–212 | MR | Zbl

[10] Liardet P., “Regularities of distribution”, Compositio Math., 61 (1987), 267–293 | MR | Zbl

[11] Petersen K., “On a series of cosecants related to a problem in ergodic theory”, Compositio Math., 26 (1973), 313–317 | MR | Zbl

[12] Rauzy G., “Nombres algebriques et substitutions”, Bull. Soc. Math. France, 110 (1982), 147–148 | MR

[13] Sierpinski W., “Sur la valeur asymptotique d'une certaine somme”, Bull Intl. Acad. Polonmaise des Sci. et des Lettres (Cracovie) series A, 1910, 9–11 | Zbl

[14] Szüsz R., “Über die Verteilung der Vielfachen einer Komplexen Zahl nach dem Modul des Einheitsquadrats”, Acta Math. Acad. Sci. Hungar., 5 (1954), 35–39 | DOI | MR | Zbl

[15] Weyl H., “Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene”, Rendicontidel Circolo Mathematico di Palermo, 30 (1910), 377–407 | DOI | Zbl

[16] Weyl H., “Über die Gleichverteilung von Zahlen mod. Eins”, Math. Ann., 77:3 (1916), 313–352 | DOI | MR | Zbl

[17] Abrosimova A. A., “BR-mnozhestva”, Chebyshevskii sbornik, 16 (2015), 8–22 (Russian) | Zbl

[18] Abrosimova A. A., “Mnozhestva ogranichennogo ostatka na dvumernom tore”, Chebyshevskii sbornik, 12:4 (2011), 15–23 (Russian) | MR

[19] Abrosimova A. A., “Srednie znachenija otklonenij dlja raspredelenija tochek na tore”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Matematika. Fizika, 5(124):26 (2012), 5–11 (Russian)

[20] Zhuravlev V. G., “Geometrizacija teoremy Gekke”, Chebyshevskii sbornik, 11:1 (2010), 125–144 (Russian) | MR

[21] Zhuravlev V. G., “Multidimensional Hecke theorem on the distribution of fractional parts”, St. Petersburg Mathematical Journal, 24:1 (2013), 71–97 | DOI | MR | Zbl

[22] Zhuravlev V. G., “Bounded remainder polyhedra”, Proceedings of the Steklov Institute of Mathematics, 280, no. 2, 2013, S71–S90 | DOI | MR | Zbl

[23] Zhuravlev V. G., “Exchanged toric developments and bounded remainder sets”, Journal of Mathematical Sciences (New York), 184:6 (2012), 716–745 | DOI | MR | Zbl

[24] Krasil'shhikov 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), 59–67 | DOI | DOI | MR | Zbl

[25] Shutov A. V., “Dvumernaja problema Gekke–Kestena”, Chebyshevskii sbornik, 12:2(38) (2011), 151–162 (Russian) | MR | Zbl

[26] Shutov A. V., “Mnogomernye obobshhenija summ drobnyh dolej i ih teoretiko-chislovye prilozhenija”, Chebyshevskii sbornik, 14:1(45) (2013), 104–118 (Russian) | MR

[27] Shutov A. V., “O minimal'nyh sistemah schislenija”, Issledovanija po algebre, teorii chisel, funkcional'nomu analizu i smezhnym voprosam, 4, Saratov, 2007, 125–138 (Russian) | MR | Zbl

[28] Shutov A. V., “O skorosti dostizhenija tochnyh granic ostatochnogo chlena v probleme Gekke–Kestena”, Chebyshevskii sbornik, 14:2(46) (2013), 173–179 (Russian)

[29] Shutov A. V., “Ob odnom semejstve dvumernyh mnozhestv ogranichennogo ostatka”, Chebyshevskii sbornik, 12:4 (2011), 264–271 (Russian) | MR | Zbl

[30] Shutov A. V., “Optimal'nye ocenki v probleme raspredelenija drobnyh dolej na mnozhestvah ogranichennogo ostatka”, Vestnik SamGU. Estestvennonauchnaja serija, 2007, no. 7 (57), 168–175 (Russian)

[31] Shutov A. V., “Raspredelenie drobnyh dolej linejnoj funkcii na mnozhestvah polozhitel'noj korazmernosti”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Matematika. Fizika, 19(162):32 (2013), 134–143 (Russian) | MR