Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type $$\sum_{x_{1} \in A_{1},\ldots, x_{r} \in A_{r}} e_{p}(x_{1},\ldots x_{r})$$ under nearly optimal conditions on $$|A_{1}| \cdots |A_{r}|$$ . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of $${\mathbb{F}_{p}^{*}}$$ , obtaining a nontrivial estimate provided $$log |H| > C {\frac{log p}{log log p}}$$ . This is a further improvement on [BGK].