  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.
