On the Strict Majorant Property in Arbitrary Dimensions

Abstract

Abstract In this work we study d-dimensional majorant properties. We prove that a set of frequencies in $\mathbb{Z}^d$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all p > 0 if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least d + 2 frequencies in $\mathbb{Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $p \not\in 2\mathbb{N}$ of length 2. Any infinite set of frequencies in $\mathbb{Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $p \not\in 2\mathbb{N}$ of length 2. Finally, given any p > 0 with $p \not\in 2\mathbb{N}$, we exhibit a set of d + 2 frequencies on the moment curve in $\mathbb{R}^d$ that violate the strict majorant property on $L^p([0,1]^d).$