On the cardinality of lower sets and universal discretization

TL;DR

This paper investigates the size of lower sets in multi-dimensional integer lattices and applies these findings to improve universal discretization methods for the $L_2$-norm of trigonometric polynomial subspaces.

Contribution

It derives new bounds on the cardinality of lower sets and utilizes these results to enhance discretization techniques for high-dimensional polynomial spaces.

Findings

New bounds on the cardinality of lower sets in $ extbf{Z}_+^d$

Refined results for lower set sizes of $n$ elements

Improved universal discretization methods for $L_2$-norms

Abstract

A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets.