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 $\limsup_{N \to \infty} N^{-1}|\mathbf B \cap [1,N]|= 0,$ for which the Hardy--Littlewood majorant property holds: $\sup_{|a_n|\le 1} \Big\| \sum_{n\in\mathbf B\cap[1, N]} a_n e^{2 πi n ξ}\Big \|_{L^p(\mathbb{T}, {\mathrm d} ξ)} \leq \mathbf{C}_p \Big\| \sum_{n\in\mathbf B\cap[1, N]} e^{2 πi n ξ} \Big\|_{L^p(\mathbb{T}, {\mathrm d} ξ)},$ 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 $|a_n| \leq 1$.

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