Uniform distribution of sequences: Transcendental number and U.D. mod 1

2018-07-26Ever since the Weyl criterion was introduced, many mathematicians tried to answer the question: Is there a transcendental number a > 1 such that aⁿ is uniform distribution (mod 1) or not. In 1964, Woody Dudley proved that though cosnθ is not uniformly distributed (mod 1) for almost all θ, f(n)cosnθ is, where f is any function such that f(n) goes to infinity with n, no matter how slowly. ❧ In this thesis, the Weyl criterion is used to figure out a transcendental number a > 1 such that aⁿ is uniform distributed (mod 1) or not. To help find such a transcendental number, a method called Euler summation will be introduced as a key to this problem. The transcendental number e will be used as an example in this thesis