Multiplicative difference sets via additive characters

We use Fourier analysis on the additive group of F-2m to give an alternativ e proof of the recent theorem of Maschietti and to prove recent conjectures of No, Chung and Yun and No, Golomb, Gong, Lee and Gaal on difference sets in the multiplicative group of F-2m, m odd. Along the way we prove a stron ger form of a celebrated theorem of Welch on the 3-valued cross-correlation of maximal length sequences.