On the cardinality of lower sets and universal discretization

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.