  We consider sums of oscillating functions on intervals in cyclic groups of
size close to the square root of the size of the group. We first prove
non-trivial estimates for intervals of length slightly larger than this square
root (bridging the "Poly\'a-Vinogradov gap" in some sense) for bounded
functions with bounded Fourier transforms. We then prove that the existence of
non-trivial estimates for ranges slightly below the square-root bound is stable
under the discrete Fourier transform, and we give applications related to trace
functions over finite fields.
