  Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability
space $(\Omega, \mu)$ spanned by a uniformly bounded Riesz basis $\Phi_N$.
Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain
universal discretization for integral norms $L_q(\Omega,\mu)$ of functions from
the collection of all subspaces of $X_N$ spanned by $v$ elements of $\Phi_N$
with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$.
  This last bound on $m$ is much better than previously known bounds which are
quadratic in $v$. Our proof uses a conditional theorem on universal sampling
discretization, and an inequality of entropy numbers in terms of greedy
approximation with respect to dictionaries.
