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# 1964-15 - ON SOME APPLICATIONS OF PROBABILITY TO AND NUMBER...

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ON SOME APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY P . ERDÖS 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 a k = oo . Piranian k=1 asked me for what sequences of integers n 1 < n 2 < . . . does there exist a power series a k zn k which converges uniformly in I z < 1 but for which k=1 z Z I a k = 00 . I proved [2] by probabilistic methods that if the sequence k=1 {n 1 .} satisfies lim inf (n ;-n 1 ) 1 i ( i - 0 = 1 where j-i-->oo then if Y a, . z111-- converges for k--1 z (1) then such a power series exists . Zygmund [3] proved that if n k+1/n k > 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 {n k} for which there exists an absolute constant c so that for every i < j n ;-n ti > ( 1+c)' - ti . (2) Curiously enough, (1) occurred in a seemingly different context . Gaier and Meyer-König [4] call the radius defined by z = re 2 O, 0 < it < 1, U) singular for f (z) = Z a k z n if f (z) is unbounded in every sector I z I < 1, 71=1 Co ~- E < arg z < 0-f- E where E > 0 . They showed that if f (z) _ a,, znk k=1 and n k+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), 692-696]

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APPLICATIONS OF PROBABILITY TO ANALYSIS AND NUMBER THEORY 693 is a singular radius . Rényi and I [5] showed by probabilistic methods W that if {n, ;} satisfies (1) then there exists a power series f (z) = E a k znk k=1 m for which a k > 0, a k = oo, thus the positive real axis is a singular k=7 radius but no other radius is singular . In fact, f (z) is bounded in I z < 1 if a region I z-1 < 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 Meyer-Köng remains true, but we
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1964-15 - ON SOME APPLICATIONS OF PROBABILITY TO AND NUMBER...

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