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 $\mathbb{ Q}\cap (0,1)\subset E$.
Categories: math.NT