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 "Polya-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. We then give applications related to trace functions over finite fields.