Character Sum and Fourier Analytic Methods: r-Critical Pairs I

Abstract

In this chapter, we give a partial extension of the 3k−4 Theorem valid for prime order groups—both for symmetric sumsets A+A as well as the distinct summand case A+B. We will use the method of Character Sums, which, though we will avoid the terminology, can be viewed in the broader context of finite Fourier Analysis. We begin with some simple lemmas about character sums.