On Some Applications of Probability To Analysis and Number Theory
P. Erdös · Journal of the London Mathematical Society · 1964
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 years. Hardy was the first to give an example of a power series E a,,, z"k k=1 which converges uniformly in z a oo (1) then such a power series exists. Zygmund [3] proved that if n k+1 /n k> c> 1 then if Y a,. z111-- converges for z 1, Z I a, ( 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 ak z n if f (z) is unbounded in every sector I z I < 1,