  Let $A\subset\mathbb{N}$, $\alpha\in(0,1)$, and for $x\in\mathbb{R}$ let
$e(x):=e^{2\pi ix}$. We set $$S_{A}(\alpha,N):=\sum_{\substack{n\in A\n\leq
N}}e(n\alpha).$$ Recently, Lambert A'Campo proposed the following question: is
there an infinite non-cofinite set $A\subset\mathbb{N}$ 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 $A$ such that $|S_{A}(\alpha,N)|$ is bounded for all
$\alpha\in E\subset (0,1)$, where $E$ has full Hausdorff dimension and
$\mathbb{Q}\cap (0,1)\subset E$.
