Short incomplete Gauss sums and rational points on metaplectic horocycles

TL;DR

This paper studies the asymptotic behavior of short incomplete Gauss sums at random arguments, revealing a limit distribution related to theta sums and employing equidistribution on metaplectic horocycles.

Contribution

It introduces a new limit law for short Gauss sums differing from long sum laws, using equidistribution on metaplectic horocycles to establish the result.

Findings

Limit distribution of short Gauss sums matches theta sums.

Difference from the law for long Gauss sums.

Utilizes equidistribution theorem on rational points in metaplectic cover.

Abstract

In the present paper we investigate the limiting behaviour of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2,R).