Finite dimensional distributions for short incomplete Gauss sums

Abstract

This paper investigates an equivalent principle to the weak invariance principle, with a focus on short incomplete Gauss sums. We establish a limit law for the finite-dimensional distributions (FDD) of these sums as the size parameter grows. Additionally, the study extends these findings to the limiting distribution of theta functions, building upon prior research by the author. This connection highlights the broader implications of the results in the context of homogeneous dynamics and modular forms.