An estimate of Gaussian sums (Q1219910)

By means of some complicated elementary considerations the author proves the inequality \[ \left|\sum_{k=0}^{q-1} e(\tfrac aq k^n)\right|\leq \exp\left(c\left(\tfrac {n}{\varphi(n)}\right)^2\right) \] uniformly in \(n,a\), and \(q\) \((n\geq 3,\;(a,q)=1)\). The importance of this result lies in the weak dependence of the upper bound on the parameter \(n\).