Classification theorems for sumsets modulo a prime

TL;DR

This paper establishes classification theorems for sumsets in finite fields of prime order, addressing when zero or all elements can be expressed as sums of subsequence elements, including sums of fixed length.

Contribution

It provides new classification results characterizing the conditions under which certain sumset representations are possible in prime order finite fields.

Findings

Characterization of when zero can be expressed as a sum of elements from A

Conditions for representing every element of Z/pZ as a sum of A's elements

Results on sums of exactly l elements from A

Abstract

Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can one represent every element of $\Z/pZ$ as a sum of some elements of $A$ ? (3) When can one represent every element of $\Z/pZ$ as a sum of $l$ elements of $A$ ?