  The aim of this paper is to exhibit a wide class of sparse deterministic
sets, $\mathbf B \subseteq \mathbb{N}$, so that
  \[ \limsup_{N \to \infty} N^{-1}|\mathbf B \cap [1,N]|= 0, \]
  for which the Hardy--Littlewood majorant property holds:
  \[ \sup_{|a_n|\le 1} \Big\| \sum_{n\in\mathbf B\cap[1, N]} a_n e^{2 \pi i n
\xi}\Big \|_{L^p(\mathbb{T}, {\mathrm d} \xi)} \leq \mathbf{C}_p \Big\|
\sum_{n\in\mathbf B\cap[1, N]} e^{2 \pi i n \xi} \Big\|_{L^p(\mathbb{T},
  {\mathrm d} \xi)}, \] where $p \geq p_{\mathbf{B}}$ is sufficiently large,
the implicit constant $\mathbf{C}_p$ is independent of $N$, and the supremum is
taken over all complex sequences $ (a_n : n \in \mathbb{N})$ such that $|a_n|
\leq 1$.
