Fouvry, EtienneKowalski, EmmanuelMichel, PhilippeRaju, Chandra SekharRivat, JoelSoundararajan, Kannan2017-03-272017-03-272017-03-27201710.5802/aif.3087https://infoscience.epfl.ch/handle/20.500.14299/135952WOS:000393926100013We 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.Short exponential sumstrace functionsvan der Corput lemmacompletion methodRiemann Hypothesis over finite fieldstext::journal::journal article::research article