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The Borel property for simple Riesz means. (English) Zbl 0597.10051
Let $$P=(p_ n)$$ be a sequence of positive real numbers. The complex sequence $$(\alpha_ n)$$ is called P-summable, if $$(1/P_ N)\sum^{N}_{n=1}p_ n\alpha_ n$$ converges, where $$P_ N=\sum^{N}_{n=1}p_ n$$. Assuming that P is increasing, it is shown that almost all sequences (with respect to any probability measure on $${\mathbb{C}})$$ are P-summable iff ”Hill’s condition” (H) holds, i.e. $$\sum^{\infty}_{n=1}\exp (-\delta /a_ n)<\infty$$ for all $$\delta >0$$, where $$a_ n=P_ n^{-2}\sum^{n}_{k=1}p^ 2_ k$$. The sufficiency of (H) is due to J. D. Hill [Pac. J. Math. 1, 399-409 (1951; Zbl 0043.286)]. For arbitrary (non-monotonic) weights, (H) is still sufficient, but not necessary. We give a related necessary and sufficient condition.
##### MSC:
 11K06 General theory of distribution modulo $$1$$ 40D09 Structure of summability fields
##### Keywords:
simple Riesz means; Borel property; uniform distribution
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##### References:
 [1] [Hi1]Hill, J. D.: The Borel property of summability methods. Pacific J.1, 399-409 (1951). · Zbl 0043.28603 [2] [Hi2]Hill, J. D.: Remarks on the Borel property. Pacific J.4, 227-242 (1954). · Zbl 0057.29301 [3] [Hl1]Hlawka, E.: Folgen auf kompakten Räumen. Abh. Math. Sem. Hamburg20, 223-241 (1956). · Zbl 0072.05701 [4] [Hl2]Hlawka, E.: Theorie der Gleichverteilung. Mannheim: B. I. 1979. · Zbl 0406.10001 [5] [KN]Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences, New York: Wiley. 1974. · Zbl 0281.10001 [6] [L]Loève, M.: Probability Theory I. 4th Ed. New York-Heidelberg-Berlin: Springer. 1977. [7] [M]Martikainen, A. I.: On necessary and sufficient conditions for the strong law of large numbers. Theory Probab. Appl.24, 813-820 (1980). · Zbl 0441.60026 [8] [SZ]Salem, R., Zygmund, A.: Some properties of trigonometric series whose terms have random signs. Acta Math.91, 245-301 (1954). · Zbl 0056.29001 [9] [T]Tsuji, M.: On the uniform distribution of numbers mod 1. J. Math. Soc. Japan4, 313-322 (1952). · Zbl 0048.03302 [10] [ZB]Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin-Heidelberg-New York: Springer. 1970. · Zbl 0199.11301
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