  For a polynomial $P$ mapping the integers into the integers, define an
averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on
functions on the integers. We prove sufficient conditions for the
$\ell^{p}$-improving inequality \begin{equation*} \|A_N
f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})}
\|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where
$1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the
inequalities established are sharp, up to the boundary of the allowed pairs of
$(p,q)$. For degree three and higher, the inequalities are close to being
sharp. In the quadratic case, we appeal to discrete fractional integrals as
studied by Stein and Wainger. In the higher degree case, we appeal to the
Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and
Guth.

Classical Analysis and ODEs