On the fourth derivative test for exponential sums

Abstract We give an upper bound for the exponential sum ∑ m=1,...,M exp(2iπf(m)) where f is a real-valued function whose fourth derivative has the order of magnitude λ > 0 small. Van der Corput's classical bound, in terms of M and λ only, involves the exponent 1/14. We show how this exponent may be replaced by any θ < 1/12 without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.