A Note on Bounded Exponential Sums

Let $A\subset\mb{N}$, $\alpha\in(0,1)$, and $e(x):=e^{2\pi ix}$ for $x\in\mb{R}$. We set
$$S_{A}(\alpha,N):=\sum_{\substack{n\in A\\n\leq N}}e(n\alpha).$$
Recently, Lambert A'Campo posed the following question: is there an infinite non-cofinite set $A\subset\mb{N}$ such that for all $\alpha\in(0,1)$ the sum $S_{A}(\alpha,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(\alpha,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A\subset\mb{N}$ such that $|S_{A}(\alpha,N)|$ is bounded independently of $N$ for all $\alpha\in E\subset (0,1)$, where $\mb{Q}\cap (0,1)\subset E$ and $E$ has full Hausdorff dimension.