Let $\Lambda\subseteq \{1,\ldots, N\}$ and let $\{a_{n}\}_{n\in\Lambda}$ be a sequence with $|a_n|\leq 1$ for all $n$. It is easy to see that\[ \left\|\sum_{n\in\Lambda}a_{n}e{(n\theta)}\right\|_{p}\leq \left\|\sum_{n\in\Lambda}e{(n\theta)}\right\|_{p}\] for every even integer $p$. We give an example which shows that this statement can fail rather dramatically when $p$ is not an even integer. This answers in the negative a question known as the Hardy–Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.