Explorations of sum-product phenomena in fields
We study the sum-product phenomena in finite fields and in reals. The well known sum-product theorem says that under certain circumstances it is not possible for a set to have both an additive and a multiplicative structure. In this dissertation, we first study a quantitative version of the sum-product theorem, and give a stronger sum-product theorem in general fields. We also use Fourier analytic machinery to investigate the expanding property of a family of two variables polynomials. We then use Fourier analytic methods to characterize the polynomials such that when a set behaves like an arithmetic progression, then the size of its image under the polynomials is large. We also study the Erdös-Falconer distance problems in vector space over finite fields. Some analogous results in real setting are also derived.