A converse to a theorem of Gauss on Gauss sums

2024/07/24 by Jonathan W. Bober, Bober, Jonathan W., Leo Goldmakher +1
Mathematics · #11L05 (Primary) 11T24 #20C15 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:11L05 #msc:11T24 #msc:20C15

paper · pdf · doi:10.48550/arxiv.2407.16937

12 pages. Comments very welcome!

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

In this note we prove (under mild hypotheses) that f is a nontrivial character of \mathbbF_p if and only if the Fourier transform of f has magnitude 1 somewhere in \mathbbF_p^×. 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.

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