zbMATH — the first resource for mathematics

Description and exact maximum and minimum values of the remainder in the problem of the distribution of fractional parts. (English. Russian original) Zbl 1303.11078
Math. Notes 89, No. 1, 59-67 (2011); translation from Mat. Zametki 89, No. 1, 43-52 (2011).
The paper deals with the distribution of the function \(f(x)=\langle \alpha x\rangle\) where \(\langle x\rangle\) is the fractional part of the \(x.\) The aim of the paper is to investigate how much can be the deviation of this distribution from the uniform one. Let \[ r(\alpha,n,I)=\#\{i: 0\leq i< n, \langle i\alpha\rangle \in I\}-n|I| \] \[ r^+(\alpha)=\sup_{n}r(\alpha,n,I),\, r^-(\alpha)=\inf_{n}r(\alpha,n,I). \] The authors consider the intervals of the form \(I=[\delta;\delta+\langle m\alpha\rangle]\) and such that \(|r^{\pm}(\alpha)|<\infty.\) Using Hecke’s formula for \(r(\alpha,n,I)\) and investigating different properties of piecewise linear functions, the authors obtain new formula for \(r^{\pm}(\alpha)\). As a corollary they prove that \(r^{\pm}(\alpha)\) can be computed in \(O(m)\) operations.

11J71 Distribution modulo one
11K38 Irregularities of distribution, discrepancy
11K31 Special sequences
Full Text: DOI
[1] E. Hecke, ”Über analytische Funktionen und die Verteilung von Zahlen mod. eins,” Hamb. Abh. 1, 54–76 (1921). · JFM 48.0184.02 · doi:10.1007/BF02940580
[2] A. Ostrowski, ”Notiz zur Theorie der Diophantischen Approximationen und zur Theorie der linearen Diophantischen Approximationen,” Math. Miszelleren XVI. Jahresber. Deutschen Math. Ver. 39, 34–46 (1939). · JFM 56.0184.01
[3] H. Kesten, ”On a conjecture of Erdos and Sz üsz related to uniform distributionmod 1,” Acta Arithmetica 12, 193–212 (1966). · Zbl 0144.28902 · doi:10.4064/aa-12-2-193-212
[4] H. Furstenberg, H. Keynes, and L. Shapiro, ”Prime flows in topological dynamics,” Israel J. Math. 14(1), 26–38 (1973). · Zbl 0264.54030 · doi:10.1007/BF02761532
[5] I. Oren, ”Admissible functions with multiple discontinuities,” Israel J. Math. 42(4), 353–360 (1982). · Zbl 0533.28009 · doi:10.1007/BF02761417
[6] K. Petersen, ”On a series of cosecants related to a problem in ergodic theory,” CompositioMath. 26, 313–317 (1973). · Zbl 0269.10030
[7] P. Liardet, ”Regularities of distribution,” Compositio Math. 61(3), 267–293 (1987). · Zbl 0619.10053
[8] G. Rauzy, ”Ensembles à restes bornés,” in Seminaire de théorie des nombres de Bordeaux 1983/1984, Exp. No. 24 (Univ. Bordeaux I, Talence, 1984).
[9] S. Ferenczi, ”Bounded remainder sets,” Acta Arithmetica 61(4), 319–326 (1992). · Zbl 0774.11037 · doi:10.4064/aa-61-4-319-326
[10] V. Berthé and R. Tijdeman, ”Balance properties of multi-dimensional words,” Theoret. Comput. Sci. 273(1–2), 197–224 (2002). · Zbl 0997.68091 · doi:10.1016/S0304-3975(00)00441-2
[11] A. V. Shutov, ”On the distribution of fractional parts,” Chebyshevskii Sb. 5(3), 112–121 (2004). · Zbl 1144.11060
[12] A.V. Shutov, ”On the distribution of fractional parts. II,” in Studies in Algebra, Number Theory, Functional Analysis, and Related Questions: Collection of Scientific Papers (Izd. Saratovsk. Univ., Saratov, 2005), Vol. 3, pp. 146–158 [in Russian].
[13] A. V. Shutov, ”Optimal estimates in the problem of the distribution of the fractional parts n\(\alpha\) on bounded remainder sets,” Vestnik Samarsk. Gos. Univ. Estestvennonauchn. Ser., No. 7, 168–175 (2007).
[14] V. V. Krasil’shchikov and A. V. Shutov, ”Embedding of lattices in quasiperiodic lattices,” in Studies in Algebra, Number Theory, Functional Analysis, and Related Questions: Collection of Scientific Papers (Izd. Saratovsk. Univ., Saratov, 2007), Vol. 4, pp. 45–55 [in Russian].
[15] V. V. Krasil’shchikov, A. V. Shutov, and V. G. Zhuravlev, ”One-dimensional quasiperiodic tilings admitting progressions enclosure,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 3–9 (2009) [Russian Math. (Iz. VUZ) 53 (7), 1–6 (2009)]. · Zbl 1195.11084
[16] H. Weyl, ”On the uniform distribution of numbersmodulo 1,” in Selected Works.Mathematics. Theoretical Physics (Nauka, Moscow, 1984), pp. 58–93 [in Russian].
[17] C. Pinner, ”On sums of fractional parts n\(\alpha\) + \(\gamma\),” J. Number Theory 65(1), 48–73 (1997). · Zbl 0886.11045 · doi:10.1006/jnth.1997.2080
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.