  We apply Christ's method of refinements to the $\ell^p$-improving problem for
discrete averages $\mathcal{A}_N$ along polynomial curves in $\mathbb{Z}^d$.
Combined with certain elementary estimates for the number of solutions to
certain special systems of diophantine equations, we obtain some restricted
weak-type $p \to p'$ estimates for the averages $\mathcal{A}_N$ in the
subcritical regime. The dependence on $N$ of the constants here obtained is
sharp, except maybe for an $\epsilon$-loss.
