Bounded Exponential Sums with Multiplicative Coefficients

For a multiplicative function $f$ and $\alpha \in \mathbb{R}$, we study when the exponential sum $\sum\limits_{n \leq x} f(n)e^{2 \pi i n \alpha}$ can be bounded. I will motivate the study of such sums and state our main classification results, and then an overview of a few of the key ideas involved. This is based on joint work with Ihor Pylaiev.