On the cardinality of lower sets and universal discretization

F. Dai; A. Prymak; A. Shadrin; V. Temlyakov; S. Tikhonov

doi:10.1016/j.jco.2022.101726

paper

On the cardinality of lower sets and universal discretization

arXiv:2208.02113 · doi:10.1016/j.jco.2022.101726

Abstract

A set $Q$ in $Z_{+}^{d}$ is a lower set if $(k_{1}, \dots, k_{d}) \in Q$ implies $(l_{1}, \dots, l_{d}) \in Q$ whenever $0 \leq l_{i} \leq k_{i}$ for all $i$ . We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $Z_{+}^{d}$ . 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.