A converse to a theorem of Gauss on Gauss sums

Abstract

In this note we prove (under mild hypotheses) that $f$ is a nontrivial character of ${\mathbb{F}}_p$ if and only if the Fourier transform of $f$ has magnitude 1 somewhere in ${\mathbb{F}}_p^{\times }$. This implies a converse to a theorem of Gauss on the magnitude of the Gauss sum, in addition to other consequences. The common theme in all our results is that extremal behavior on the Fourier side imposes multiplicative structure on the physical side.

12 pages. Comments very welcome!