Some Subcritical Estimates for the [Formula omitted]-Improving Problem for Discrete Curves

We apply Christ's method of refinements to the [Formula omitted]-improving problem for discrete averages [Formula omitted] along polynomial curves in [Formula omitted]. Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type [Formula omitted] estimates for the averages [Formula omitted] in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an [Formula omitted]-loss.