Cauchy-Davenport Theorem for abelian groups and diagonal congruences

Abstract

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 = 1 n a i x i k ≡ c ( mod q ) \sum _{i=1}^n a_ix_i^k \equiv c \pmod q with an arbitrary modulus.