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Unformatted text preview: ON SOME APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY P . ERDS It would be quite impossible to give a survey of these subjects in a short article or lecture, and I will only succeed by making some arbitrary restrictions on the topics with which I will deal . First of all, I will restrict myself to problems and results on which I worked, and secondly, I will not discuss subjects which have been discussed in recently appeared review articles [1]. Probabilistic methods have been used in analysis for several decades ; it suffices to name Paley, Wiener, Kolmogoroff, Zygmund, Salem, Steinhaus, Kac, Dvoretzky, Kahane, and many others. I will restrict myself to some questions my collaborators and I worked on for several ar years. Hardy was the first to give an example of a power series E a,,, z " k k=1 cA which converges uniformly in z a < 1 but for which ak = oo. Piranian k=1 asked me for what sequences of integers n1< n2 < ... does there exist a power series ak zn k which converges uniformly in I z < 1 but for which k=1 z Z I ak = 00 . I proved [2] by probabilistic methods that if the sequence k=1 {n1 .} satisfies lim inf (n;  n 1) 1 i(i 0 = 1 where ji>oo then if Y a,. z111 converges for k1 z (1) then such a power series exists. Zygmund [3] proved that ifnk+1/nk> c > 1 1, Z I a, < oo . Thus (1) is certainly k=1 not far from being best possible, and it is quite likely that it is, in fact, best possible ; in fact, Zygmund's theorem may remain true for every sequence which does not satisfy (1), in other words, for every sequence {nk}for which there exists an absolute constant c so that for every i < j n;nti> (1+c)' ti. (2) Curiously enough, (1) occurred in a seemingly different context . Gaier and MeyerKnig [4] call the radius defined by z = re 2 O, 0 <it <1, U) singular for f (z) = Z ak zn if f (z) is unbounded in every sector I z I < 1, 71=1 Co ~ E < arg z <0f E where E > 0. They showed that if f (z) _ a,, znk k=1 and nk+1/n k > c > 1, and if f (z) is unbounded in (z ~ < 1, then every radius Received 1 May, 1963 . [JOURNAL LONDON MATH . Soc ., 39 (1964), 692696] APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY 693 is a singular radius. Rnyi and I [5] showed by probabilistic methods W that if {n,;}satisfies (1) then there exists a power seriesf (z)= E ak znk k=1 m for which a k > 0, ak = oo, thus the positive real axis is a singular k=7 radius but no other radius is singular. In fact, f (z) isbounded in I z < 1 if a region I z1 < e is excluded (for every e > 0). It again seems quite possible that our theorem is best possible;in fact, perhaps if{n k }satisfies (2) then the theorem of Gaier and MeyerKng remains true, but we could prove nothing in this direction.could prove nothing in this direction....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).
 Winter '11
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