On the range of ${\sum }_{n=1}^{\infty }\pm c_n$

Abstract

Let $\{c_n{\}}_{n=1}^{\infty }$ be a sequence of complex numbers. In this paper we answer when the range of ${\sum }_{n=1}^{\infty }\pm c_n$ is dense or equal to the complex plane. Some examples are given to explain our results. As its application, we calculate the Hausdorff dimension of the level sets of a Rademacher series with complex coefficients.

arXiv comment15 pages