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 F-p(x) at which the Fourier transform of a characteristic function chi(A), A subset of or equal to F-p, P 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\ = [p/2], and if R is the set of points r for which \(χ) over cap (A)(r)\ greater than or equal to alphap, then almost nothing can be said about R other than that \R\ much less than alpha(-2), a trivial consequence of Parseval's theorem.