Cauchy-Davenport Theorem for abelian groups and diagonal congruences

We prove an analogue of the Cauchy-Davenport Theorem and Chowla’s Theorem for sum sets in a general abelian group and give an application to diagonal congruences, establishing a best possible estimate for the distribution of solutions of a diagonal congruence ∑i=1naixik≡c(modq)\sum _{i=1}^n a_ix_i^k \equiv c \pmod q with an arbitrary modulus.