Miguel A. Lerma's Research Plan

During my current research I have studied several problems arising from the subject of uniform distribution modulo 1 (see my \Current Research" report). Two subjects of interest that deserve further exploration are explained below. 1. Extremal functions in Fourier Analysis Vaaler ((9]) has used Beurling's function (an extremal majorizing function for sgn(x)) to nd simple proofs for the Erdd os-Turan inequality and related results. As I have shown during my current research, analogous results can also be obtained for the conjugate function of sgn(x), i.e. log jxj. Note that except for a multiplicative constant, the functions sgn(x) and log jxj are respectively the limits of the imaginary part and the real part of log(x + yi) as y ! 0 + , hence a natural generalization of the previous results would be to nd respectively extremal majorizing (and minorizing) entire functions of exponential type 2 for the functions f y (x) = 0. One can expect the analyticity properties of log z in the upper half complex plane to play some role in the proofs. After obtaining the corresponding extremal functions, modiied versions for the Erdd os-Turan inequality and the Montgomery and Vaughan's inequality, now depending on a parameter y > 0, will be easy to obtain. By letting y ! 0 + we will recover the previous results about sgn(x) and log jxj. My purpose is to explore the mentioned generalization, and look for a way to unify the results in a single comprehensive theory in which a main theorem would have the above mentioned results as mere corollaries. The techniques developed in this study might be suitable for generalization to other functions that are analytic in the upper half plane besides log z. A family of functions that might deserve study are the polylogarithms, from which the usual logarithmic function is a particular case. 2. Computability of numbers outside the Koksma's exceptional set It is well known that, as a consequence of Koksma's Metric Theorem ((4],,5, theorem 1.4.3]), for almost every real number > 1 (in the sense of Lebesgue measure) the sequence f n g 1 n=1 is uniformly distributed modulo 1 (u.d. mod 1). However, it is also the case that nobody knows of any speciic real number for which that property can be proven 1. The present situation is similar …