The value distribution of incomplete Gauss sums

TL;DR

This paper investigates the distribution of incomplete Gauss sums, revealing a limit law that describes their value distribution when summing over subintervals, expanding understanding beyond classical Gauss sums.

Contribution

It establishes a limit law for the value distribution of incomplete Gauss sums, linking it to a family of periodic functions, and complements existing bounds and limit theorems.

Findings

Limit law for incomplete Gauss sums established

Distribution characterized by a family of periodic functions

Results extend understanding of value distribution beyond classical sums

Abstract

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.