Multiple rational trigonometric sums and multiple integrals

V. N. Chubarikov

Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 61-68.

Résumé

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$ \biggl|\sum_{x_1,\dots,x_r=1}^q\exp(2\pi if(x_1,\dots,x_r)/q)\biggr|\ll q^{r-1/n+\varepsilon} $$ where \begin{gather*} f(x_1,\dots,x_r)=\sum\nolimits_{0\le t_1,\dots,t_r\le n^at_1,\dots,t_r}x_1^{t_1}\dots x_r^{t_r}, \\ a_{0,\dots,0}=0,\quad(a_{0,\dots,0,1}\dots,a_{n,\dots,n},q)=1, \end{gather*} and an estimate of the modulus of a multiple trigonometric integral.