Exponential sums and additive combinatorics

Abstract

In this thesis we provide new results in additive combinatorics which in turn lead us to new bounds of certain exponential sums. We also use known bounds on exponential and character sums to give new results in additive combinatorics. Specifically we will see how bounds on some quantities from additive combinatorics appear naturally when trying to bound multilinear exponential sums. We then find applications to bounds of exponential sums of sparse polynomials. We also give new bounds for an analogue of the energy variant of the sum-product problem over arbitrary finite fields.