Abstract
The paper deals with local discrepancies in the problem of the distribution of the sequence \(\{k\alpha\}\), i.e., with the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. A construction of intervals for which local discrepancies tend to infinity slower than any given function is presented. Moreover, it is shown that there exists an uncountable set of such intervals. Previously, similar results were obtained only for irrationalities with bounded partial quotients of their continued fraction expansions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Bohl, “Über ein in der Theorie der säkutaren Störungen vorkommendes Problem,” J. Reine Angew. Math. 135, 189–283 (1909).
W. Sierpinski, “Sur la valeur asymptotique d’une certaine somme,” Bull. Intl. Acad. Polonmaise des Sci. et des Lettres (Cracovie). Ser. A, 9–11 (1910).
H. Weyl, “Über die Gibbs’sche Erscheinung und verwandte Konvergenzphänomene,” Rend. Circ. Math. Palermo 30, 377–407 (1910).
M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, in Lecture Notes in Math. (Springer- Verlag, Berlin, 1997), Vol. 1651.
C. G. Pinner, “On Sums of Fractional Parts \(\{n\alpha+\gamma\}\),” J. Number Theory 65 (1), 48–73 (1997).
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences (Interscience, New York– London– Sydney, 1974).
E. Hecke, “Über analytische Funktionen und die Verteilung van Zahlen mod Eins,” Abh. Math. Sem. Univ. Hamburg 5, 54–76 (1921).
H. Kesten, “On a conjecture of Erdös and Szüsz related to uniform distribution mod 1,” Acta Arith. 12 (2), 193–212 (1966).
A. V. Shutov, “Optimal estimates in the problem of distribution of fractional parts on sets of bounded remainder,” Vestn. SamGU, No. 7 (57), 168–175 (2007).
V. V. Krasil’shchikov and A. V. Shutov, “Description and exact maximum and minimum values of the remainder in the problem of the distribution of fractional parts,” Math. Notes 89 (1), 59–67 (2011).
V. T. Sós, “On strong irregularities of the distribution of \(\{n\alpha\}\) sequences,” in Studies in Pure Mathematics (Birkhäuser, Basel, 1983), pp. 685–700.
B. Adamczewski, “Repartition des suites \((n\alpha)_{n\in\mathbb{N}}\) et substitutions,” Acta Arith. 112 (1), 1–22 (2004).
L. Roçadas and J. Schoißengeier, “On the local discrepancy of \((n\alpha)\)-sequences,” J. Number Theory 131 (8), 1492–1497 (2011).
J. Beck, “Randomness of the square root of 2 and the Giant Leap. Part 1,” Period. Math. Hungar. 60 (2), 137–242 (2010).
J. Beck, “Randomness of the square root of 2 and the Giant Leap. Part 2,” Period. Math. Hungar. 62 (2), 127–246 (2011).
A. V. Shutov, “Local discrepancies in the problem of the distribution of the fractional parts of a linear function,” Russian Math. (Iz. VUZ) 61 (2), 74–82 (2017).
W. Schmidt, “Irregularities of distribution. VII,” Acta Arith. 21 (1), 45–50 (1972).
A. V. Shutov, “Inhomogeneous Diophantine approximations and distribution of fractional parts,” J. Math. Sci. 182 (4), 576–585 (2012).
S. Fukasawa, “Über die Grössenordnung des absoluten Betrages von einer linearen inhomogenen Form. I,” Japanese Journal of Mathematics 3, 1–26 (1926).
H. Davenport, “On a theorem of Khintchine,” Proc. London Math. Soc. (2) 52 (1), 65–80 (1950).
J. W. S. Cassels, “Über \(\liminf_{x\to\infty} x|\theta x+\alpha-y|\),” Math. Ann. 127 (1), 288–304 (1954).
T. W. Cusick, A. M. Rockett and P. Szüsz, “On inhomogeneous Diophantine approximation,” J. Number Theory 48 (3), 259–283 (1994).
C. G. Pinner, “More on inhomogeneous diophantine approximation,” J. Théor. Nombres Bordeaux 13 (2), 539–557 (2001).
T. Komatsu, “On inhomogeneous continued fraction expansions and inhomogeneous diophantine approximation,” J. Number Theory 62 (1), 192–212 (1997).
D. Hensley, Continued Fractions (World Sci. Publ., Hackensack, NJ, 2006).
C. D. Olds, Continued Fractions (Random House, New York, 1963).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 452-463 https://doi.org/10.4213/mzm11326.
Rights and permissions
About this article
Cite this article
Shutov, A.V. Local Discrepancies in the Problem of the Distribution of the Sequence \(\{k\alpha\}\). Math Notes 109, 473–482 (2021). https://doi.org/10.1134/S0001434621030147
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434621030147