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

Some subcritical estimates for the $\ell^p$-improving problem for discrete curves