On the Hardy-Littlewood majorant problem for arithmetic sets

The aim of this paper is to exhibit a wide class of sparse deterministic sets, B⊆N, so thatlimsupN→∞N−1|B∩[1,N]|=0, for which the Hardy-Littlewood majorant property holds:sup|an|≤1⁡‖∑n∈B∩[1,N]ane2πinξ‖Lp(T,dξ)≤Cp‖∑n∈B∩[1,N]e2πinξ‖Lp(T,dξ), where p≥pB is sufficiently large, the implicit constant Cp is independent of N, and the supremum is taken over all complex sequences (an:n∈N) such that |an|≤1.