  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.
