On the Hardy--Littlewood majorant problem for arithmetic sets

The aim of this paper is to exhibit a wide class of sparse deterministic sets, $\mathbf{B}\subseteq \mathbb{N}$, so that $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{N}^{-1}\mathrm{\mid }\mathbf{B}\cap [1,N]\mathrm{\mid }=0,$$ for which the Hardy–Littlewood majorant property holds: $$\underset{\mathrm{\mid }{a}_n\mathrm{\mid }\le 1}{\sup }\parallel \sum _{n\in \mathbf{B}\cap [1,N]}{a}_n{e}^{2\pi in\xi }{\parallel }_{L^p(\mathbb{T},\mathrm{d}\xi )}\le {\mathbf{C}}_p\parallel \sum _{n\in \mathbf{B}\cap [1,N]}{e}^{2\pi in\xi }{\parallel }_{L^p(\mathbb{T},\mathrm{d}\xi )},$$ where $p\ge p_{\mathbf{B}}$ 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 $|a_n| \leq 1$.