## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  148.05402
Autor:  Erdös, Pál
Title:  Remarks on a theorem of Zygmund (In English)
Source:  Proc. Lond. Math. Soc., III. Ser. 14 A, 81-85 (1965).
Review:  We call a sequence of integers n1 < n2 < ··· a Zygmund sequence if whenever |ak| ––> 0, the power series

sumk = 1oo ak znk

converges for at least one z with |z| = 1. It is known that any sequence {nk} satisfying nk+1/nk > 1+c (c > 0) is a Zygmund sequence, and that a Zygmund sequence con not contain arbitrarily long arithmetic progressions [cf. J.-P. Kahane (Zbl 121.30102)]. The author shows the following: Let n1 < n2 < ··· be a sequence which contains two subsequences {nki} and {nli}, 1 \leq i < oo, satisfying

ki ––> oo,   ki < li < ki+1,   li-ki ––> oo ,  (nli-nki)1/(li-ki) ––> 1.

Then the above sequences is not a Zygmund sequence.
Reviewer:  M.Kinukawa
Classif.:  * 30B10 Power series (one complex variable)
Index Words:  complex functions

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