Result: Some Constructions in the Inverse Spectral Theory of Cyclic Groups: Some constructions in the inverse spectral theory of cyclic groups

Summary: The results of this paper concern the `large spectra' of sets, by which we mean the set of points in \({\mathbb F}_p^{\times}\) at which the Fourier transform of a characteristic function \(\chi_A\), \(A\subseteq {\mathbb F}_p\), 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 \(|A|=\lfloor {p/2}\rfloor\), and if \(R\) is the set of points \(r\) for which \(|\hat{\chi}_A(r)|\geqslant \alpha p\), then almost nothing can be said about \(R\) other than that \(|R|\ll \alpha^{-2}\), a trivial consequence of Parseval's theorem.