On the cardinality of lower sets and universal discretization

TL;DR

This work analyzes the size and structure of downward-closed (lower) sets in $\mathbb{Z}_+^d$, introducing precise, explicit bounds on the number $p_d(n)$ of such sets of cardinality $n$. By combining combinatorial constructions with inductive dimension-splitting arguments, the authors prove tight uniform bounds: $\frac{1}{(n-1)!} d^{n-1} < p_d(n) \le d^{n-1}$ and derive explicit, dimension-uniform estimates for $\frac{\ln p_d(n)}{n^{1-1/d}}$ with computable constants. These results are then employed to obtain universal discretization bounds for the $L_2$-norm of functions from $n$-dimensional subspaces generated by lower-set harmonics, yielding two concrete sampling-size regimes: $m\le c n^2 \ln d$ and $m\le c n^{2-1/d} d^{\ln d}$, depending on which bound for $p_d(n)$ is applicable. The findings illuminate the trade-offs between lower-set structure and discretization efficiency, with implications for sparse multivariate approximation and universal sampling strategies in high dimensions.

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.

Theorems & Definitions (15)

• Definition 1.1: Lower set
• Definition 1.2: Positive lower set
• Definition 1.3: Integer partition in ${\mathbb R}^d$
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1
• Remark 3.2