On the Hardy--Littlewood majorant problem for arithmetic sets

Abstract

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 $p\ge p_{\mathbf{B}}$ 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 $|a_n|\le 1$.