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.