Capacity of sets and uniform distribution of sequences

Abstract

In this paper the magnitude of the set of pointsx is studied for which the series\(\frac{1}{{V(N)}}\sum\limits_{n = - N}^N {c_n e^{2\pi inx} } \), with complex numberscn and an increasing sequenceV(N) of real numbers, is unbounded. An answer in terms of capacities is given. This result is then used to obtain results about exceptional sets in the theory of uniform distribution, e.g. it is shown that the spectrum of any sequence has dimension zero.