The sliding-sum method for short exponential sums

We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are barely insufficient to obtain non-trivial results. In particular, we prove various estimates for exponential sums over intervals in finite fields and related sums just below the Polya-Vinogradov range, and derive applications to equidistribution problems.