On the strict majorant property in arbitrary dimensions

TL;DR

This paper characterizes when sets of frequencies in multi-dimensional integer lattices satisfy the strict majorant property in L^p spaces, revealing that affine independence is necessary and identifying specific violations for larger or infinite sets.

Contribution

It provides a complete characterization of the strict majorant property in arbitrary dimensions and constructs explicit examples of violations for various sets of frequencies.

Findings

Sets of affinely independent frequencies satisfy the strict majorant property for all p>0.

Any set with at least d+2 frequencies violates the property for some p outside 2N.

Infinite frequency sets violate the property on infinitely many p intervals.

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 $L^p([0,1]^d).$