Trigonometric series convergent to $0$ a.e.?

Question

I was wondering whether there exist $(c_k)_{k\in\mathbb{Z}}$ in $\mathbb{C},$ not all $0,$ so that $\lim_{K \to \infty} \sum_{|k| \le K} c_ke^{2\pi i kx} = 0$ for a.e. $x \in [0,1]$. Of course the answer is "no" provided that $\sum_k |c_k|^2 < \infty$. This is all I got.

Answer 1

Yes, such series exist. They can be obtained by letting $c_k = \int_0^1 e^{-2\pi i k x}d\mu(x)$ for certain singular measures $\mu$ (sometimes called Menshov measures). Most Cantor-type measures work for this purpose, although the standard middle-third Cantor measure does not.

A set $E\subset [0,1]$ is a set of uniqueness if the only trigonometric series that converges to $0$ on $[0,1]\setminus E$ is the series of zero coefficients. Your question amounts to asking whether there exists a set of measure zero that is not a set of uniqueness. The affirmative answer was given by Menshov in 1916. The Wikipedia article on sets of uniqueness outlines the history of the subject, gives references, and adds the following curious fact:

Salem and Zygmund showed that a Cantor-like set with dissection ratio $\xi$ is a set of uniqueness if and only if $1/\xi$ is a Pisot number, that is an algebraic integer with the property that all its conjugates (if any) are smaller than $1$.