From oscillatory integrals to complete exponential sums

In \cite{PS-I}, Phong and Stein establish a sharp and stable bound for (one dimensional) scalar oscillatory integrals with a polynomial phase $\phi$ in terms of root clusters of the derivative $\phi'$. In this note we prove an analogous result for complete exponential sums. When one considers only singleton clusters, the corresponding estimate for exponential sums was established by Loxton and Vaughan in \cite{LV}. Considering all possible clusters containing a particular root allows one to obtain bounds for exponential sums which are stable under perturbations of the phase.