On the typical growth of Gaussian exponential sums

In this talk, we discuss exponential sums of the form $S(N,a,b)=\sum_{0\le n\le N-1} e^{-2\pi i (a n^2/2+nb)}$, where $a$ and $b$ are real parameters. We study its behavior for $N\to\infty$. One of the results is that for any non-decreasing function $g:\mathbb{R}_+\to\mathbb{R}_+$, for almost all $(a,b)\in (0,1)\times (-1/2,1/2]$, the limit $\limsup_{N\to+\infty}\left(g(\ln N)\,\frac{|S(N,a,b)\,|}{\sqrt{N}}\right)$ is finite if and only if $ \sum_{l\ge 1}g^6(l)<0$. The results described in the talk are obtained in collaboration with Frederic Klopp (University Paris 13).