Bounded Exponential Sums with Multiplicative Coefficients

We investigate when the exponential sum ∑n ≤ xf(n)e2 πi n α is bounded, for a multiplicative function f and α∈ R. We show that under certain natural assumptions, this exponential sum is bounded only when f closely resembles a periodic function. For completely multiplicative functions or those taking only finitely many values, we obtain sharper classification results, including a full classification (up to a measure-zero set of $α$) in the completely multiplicative case.