×

Some estimates of the \(H_{\infty}\)-uniform distribution. (English) Zbl 0624.10040

The sequences \((y_ n)=(a+b \log n+\epsilon_ n)\) are uniformly distributed in the sense of the summation method \(H_{\infty}\). In the present paper the speed of convergence of this procedure is estimated for these sequences and for some other sequences. For the sequences \((a+b \log n)\) the \(H_{\infty}\)-means converge considerably faster than logarithmic means.

MSC:

11K06 General theory of distribution modulo \(1\)
40G99 Special methods of summability
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Hlawka, E.: Theorie der Gleichverteilung. Mannheim-Wien-Zürich: B.I. 1979.
[2] Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. New York: J. Wiley. 1974. · Zbl 0281.10001
[3] Nagasaka, K.: On Benford’s law. Ann. Inst. Statist. Math.36, 337-352 (1984). · Zbl 0555.62016 · doi:10.1007/BF02481974
[4] Schatte, P.: Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgröße. Z. Angew. Math. und Mech.53, 553-565 (1973). · Zbl 0267.60025 · doi:10.1002/zamm.19730530807
[5] Schatte, P.: DieH ?-Limitierbarkeit einer Klasse von Zahlenfolgen. Math. Nachr.60, 181-190 (1974). · Zbl 0273.40007 · doi:10.1002/mana.19740600118
[6] Schatte, P.: Ein Kriterium für dieH ?-Limitierbarkeit. Math. Nachr.64, 63-70 (1974). · Zbl 0302.40010 · doi:10.1002/mana.19740640105
[7] Schatte, P.: OnH ?-summability and the uniform distribution of sequences. Math. Nachr.113, 237-243 (1983). · Zbl 0526.10043 · doi:10.1002/mana.19831130122
[8] Schatte, P.: On the asymptotic logarithmic distribution of the floating-point mantissas of sums. Math. Nachr.127, 7-20 (1986). · Zbl 0607.60022 · doi:10.1002/mana.19861270102
[9] Schatte, P.: On the almost sure convergence of floating-point mantissas and Benford’s law. Math. Nachr. To appear 1987. · Zbl 0645.60038
[10] Schatte, P.: The asymptotic behaviour of the mantissa distributions of sums. J. Inf. Process. Cybern. EIK23 (1987). · Zbl 0637.60032
[11] Tichy, R. F.: Uniform distribution and diophantine inequalities. Mh. Math.99, 147-152 (1985). · Zbl 0551.10040 · doi:10.1007/BF01304194
[12] Tichy, R. F.: Gleichverteilung zum SummierungsverfahrenH ?. Math. Nachr. To appear 1987.
[13] Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin-Heidelberg-New York: Springer. 1970. · Zbl 0199.11301
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.