On the Hardy--Littlewood majorant problem for arithmetic sets

TL;DR

This paper identifies a broad class of sparse deterministic sets for which the Hardy--Littlewood majorant property holds, extending understanding of Fourier analysis on sparse sets with applications to harmonic analysis.

Contribution

It demonstrates that certain sparse deterministic sets with zero density satisfy the Hardy--Littlewood majorant property for sufficiently large p, broadening the scope of known sets with this property.

Findings

Hardy--Littlewood majorant property holds for a wide class of sparse sets

The property is valid for sets with zero natural density

The result applies for large enough p, independent of N

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: \[ \sup_{|a_n|\le 1} \Big\| \sum_{n\in\mathbf B\cap[1, N]} a_n e^{2 \pi i n \xi}\Big \|_{L^p(\mathbb{T}, {\mathrm d} \xi)} \leq \mathbf{C}_p \Big\| \sum_{n\in\mathbf B\cap[1, N]} e^{2 \pi i n \xi} \Big\|_{L^p(\mathbb{T}, {\mathrm d} \xi)}, \] 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 $|a_n| \leq 1$.