Some Constructions In The Inverse Spectral Theory Of Cyclic Groups

The results of this paper concern the “large spectra” of sets, by which we mean the set of points in Fp at which the Fourier transform of a characteristic function χA, A ⊆ Fp, can be large. We show that a recent result of Chang concerning the structure of the large spectrum is best possible. Chang’s result has already found a number of applications in combinatorial number theory. We also show that if |A| = bp/2c, and if R is the set of points r for which |χ̂A(r)| ≥ αp, then almost nothing can be said about R other than that |R| α−2, a trivial consequence of Parseval’s theorem.