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Abstract.  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]. [ABSTRACT FROM AUTHOR]

Abstract.  In this paper, we address the problem of explicit construction of so-called ‘affine extractors’ of δ-entropy ratio, when δ ≤ 1/2 (for δ > 1/2, there is a well-known and easy example based on the Hadamard function). We first give examples for δ slightly below 1/2 and produce then a construction for arbitrary given δ > 0. All these examples are again of a simple algebraic nature. The mathematics involved includes recent developments in the sum-product theory in finite fields and certain new bounds on multilinear exponential sums. [ABSTRACT FROM AUTHOR]