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The distribution of the values of a random power series in the unit disk

Published online by Cambridge University Press:  14 November 2011

Matthias Jakob  and
A. C. Offord
Matthias Jakob
Affiliation:
Potsdamerstr. 168, 1000 Berlin 30, B.R.D
A. C. Offord
Affiliation:
Imperial College of Science and Technology, London S.W.7
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Synopsis

This is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that if

then almost all these power series take every complex value infinitely often in the unit disk.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 251 - 263
Copyright
Copyright © Royal Society of Edinburgh 1983

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References

1.Erdös, P.. On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 (1945), 898–902.CrossRefGoogle Scholar
2.Esseen, C. G.. Fourier Analysis of Distribution Functions. Acta Math. 77 (1945), 1–l25.CrossRefGoogle Scholar
3.Hayman, W. K.. Meromorphic Functions (Oxford: Univ. Press, 1964).Google Scholar
4.Jakob, M.. Nullstellen zufälliger Pötenzreiher in Einkeitskreis Dissertation an der Technischen Universität, Berlin, 1978.Google Scholar
5.Katona, G.. On a conjecture of Erdos and a stronger form of Sperner's theorem. Studia Sci. Math. Hungar. 1 (1966), 59–63.Google Scholar
6.Kleitman, D. J.. On a lemma of Littlewood and Offord on the distribution of certain sums. Math. Z 90 (1965), 251–259.CrossRefGoogle Scholar
7.Kleitman, D. J.. On a lemma of Littlewood and Offord on the distribution of linear combinations of vectors. Adv. in Math. 5 (1970), 155–157.CrossRefGoogle Scholar
8.Littlewood, J. E. and Offord, A. C.. On the number of real roots of a random algebraic equation (111). Mat. Sb. 12 (1943), 277–285.Google Scholar
9.Murai, T.. Sur la distribution des valeurs des séries lacunaires. J. London Math. Soc. 21 (1980), 93–110.CrossRefGoogle Scholar
10.Murai, T.. Value distribution of Random Taylor series in the unit disk. J. London Math. Soc. 24 (1981), 480–494.CrossRefGoogle Scholar
11.Offord, A. C.. The distribution of zeros of power series whose coefficients are independent random variables. Indian J. Math. 9 (1967), 175–196.Google Scholar
12.Offord, A. C.. The distribution of the values of a random function in the unit disk. Studia Math. 41 (1972), 71–106.CrossRefGoogle Scholar
13.A, R. E.. Paley, C. and Zygmund, A.. Some series of functions (3). Proc. Canxb. Philos. Soc. 28 (1932), 190–205.Google Scholar