The present paper is devoted to sum sets in a general abelian group, to an application to diagonal congruences, and deals with generalizations of the Cauchy-Davenport theorem.
One can note the following authors’ description of this research:
“The most significant result of this paper is a best possible estimate on the uniform distribution of points satisfying a diagonal congruence \[ \sum^{n} _{i=1}{a_ix^k _i}\equiv c\pmod q \] with an arbitrary modulus. In order to prove the result we needed an appropriate generalization of the Cauchy-Davenport Theorem to a general modulus that went beyond what Chowla’s Theorem provided. This led us to a variation of Kneser’s Theorem for abelian groups that we have not been able to find in the literature and that we believe has interest in its own right. ”
The special attention is given to auxiliary notions, examples, and statements. For example, the Cauchy-Davenport theorem, Chowla’s extension, and Olson’s estimate, are considered.
A generalization of Olson’s result for the case of abelian groups and a generalization of Chowla’s theorem to abelian groups are given. The main results of this paper are explained. Also, a refinement of Olson’s estimate is given.