Abstract
For an irrational \(\alpha \), we investigate the sums \(\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) \) and \(\sum _{i=1}^n \left\{ \left( \{i \alpha \} - \frac{1}{2} \right) ^2 - \frac{1}{12} \right\} \). We discuss exact formulae and asymptotic estimates for these sums and point out interesting geometrical properties of their graphs in the case when the continued fraction expansion of \(\alpha \) has a large isolated partial quotient.
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References
A. Bazarova, I. Berkes, L. Horváth, On the extremal theory of continued fractions. J. Theor. Prob. 29, 248–266 (2016)
J. Beck, Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting (Springer, NewYork, 2014)
H. Behnke, Über die Verteilung von Irrationalitaten mod. 1. Abh. Math. Sem. Univ. Hamburg 1, 252–267 (1922)
K. Doi, N. Shimaru, K. Takashima, On upper estimates for discrepancies of irrational rotations: via rational rotation approximations. Acta Math. Hungr. 152(1), 109–113 (2017)
M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math. 1651, Springer, Berlin (1997)
G.H. Hardy, J.E. Littlewood, Some problems of Diophantine approximation: the lattice-points of a right-angled triangle. Abh. Math. Sem. Univ. Hamburg 1, 212–249 (1922)
G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Clarendon Press, Oxford, 1979)
E. Hecke, Über analytische Functionen und die Verteilung von Zahlen mod. Eins, Abh. Math. Sem. Univ. Hamburg 1, 54–76 (1922)
M. Iosifescu, C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and its Applications, 547 (Kluwer Academic Pub, Dordrecht, 2002)
A. Khintchine, Ein Satz über Kettenbruche, mit arithmetischen Anwendungen. Ann. Zeit. 18, 289–306 (1923)
A. Khintchine, Einige Sätze über Ketterbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92, 115–125 (1924)
A. Khinchine, Metrische Kettenbruchprobleme. Compositio Math. 1, 361–382 (1935)
A.Ya. Khinchin, Continued Fractions (Dover Publications, New York, 1997)
L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974)
Y. Mori, K. Takashima, On the distribution of leading digits of \(a^n\): a study via \(\chi ^2\) statistics. Periodica Math. Hungr. 73, 224–239 (2016)
H. Niederreiter, Application of Diophantine Approximations to Numerical Integration, in Diophantine Approximation and its Applications, ed. by C.F. Osgood (Academic Press, New York, 1973), pp. 129–199
A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen. Abh. Math. Sem. Univ. Hamburg 1, 77–98 (1922)
J. Schoissengeier, On the discrepancy of \((n \alpha )\). Acta Arith. 44, 241–279 (1984)
J. Schoissengeier, On the discrepancy of \((n \alpha )\), II. J. Number Theor. 24, 54–64 (1986)
J. Schoissengeier, Eine explizite Formel für \(\sum _{n \ge N} B_2 (\{n \alpha \} )\), L.N.M. Springer, 1262, 134–138 (1987)
T. Setokuchi, On the discrepancy of irrational rotations with isolated large partial quotients: long term effects. Acta Math. Hungr. 147(2), 368–385 (2015)
T. Setokuchi, K. Takashima, Discrepancies of irrational rotations with isolated large partial quotient. Unif. Distrib. Theory 2(9), 31–57 (2014)
N. Shimaru, K. Takashima, On Discrepancies of Irrational Rotations: An Approach via Rational Rotations, Periodica Math. Hungr. (2016)
N. Shimaru, K. Takashima, Continued Fractions and Irrational Rotations. Acta Math. Hungr. 156(2), 449–458 (2016)
J. Vinson, Partial Sums of \(\zeta (\frac{1}{2})\) Modulo 1. Exp. Math. 10, 337–344 (2001)
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Mori, Y., Shimaru, N. & Takashima, K. On the distribution of partial sums of irrational rotations. Period Math Hung 78, 88–97 (2019). https://doi.org/10.1007/s10998-018-00273-y
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DOI: https://doi.org/10.1007/s10998-018-00273-y