Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-19T07:19:30.831Z Has data issue: false hasContentIssue false

Some Constructions in the Inverse Spectral Theory of Cyclic Groups

Published online by Cambridge University Press:  17 March 2003

BEN GREEN
Affiliation:
Trinity College, Cambridge CB2 1TQ, England (e-mail: Ben Green bjg23@hermes.cam.ac.uk)

Abstract

The results of this paper concern the ‘large spectra’ of sets, by which we mean the set of points in ${\bb F}_p^{\times}$ at which the Fourier transform of a characteristic function $\chi_A$, $A\subseteq {\bb F}_p$, 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.

Type
Research Article
Copyright
2003 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)