Multilinear exponential sums in prime fields under optimal entropy condition on the sources (Q1005069)

The main result of this article is to estimate \[ \bigg|\sum\limits_{x_1\in A_1,\ldots\,,x_r\in A_r}e_p(x_1\ldots\,x_r)\bigg|<p^{-\delta '}|A_1|\ldots |A_r|\,, \] where \(A_i\subset \mathbb{F}_p,\;|A_i|>p^\delta ,\,i=1,\ldots ,r,\,0<\delta <1/4,\,\prod\limits_{i=1}^r|A_i|>p^{1+\delta}\) and \(\delta '>\big(\frac{\delta}{r}\big)^{C_r}\). Taking \(A_1=\ldots =A_r=H\), where \(H\) is a multiplicative subgroup of \(\mathbb{F}^*_p\) the upper bound \[ \max\limits_{(a,p)=1}\bigg|\sum\limits_{x\in H}e_p(ax)\bigg|<p^{-\delta '}|H|, \] \(\delta '>\exp(-C/\delta),\,C>1\) is obtained. It gives a nontrivial estimate provided \(\log |H|>C'\frac{\log p}{\log\log p},\,C'>1\). This is an improvement of a result in the former work of \textit{J. Bourgain, A. A. Glibichuk} and \textit{S. V. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)].