A generalization of sumsets modulo a prime

TL;DR

This paper extends the understanding of generalized sumsets by establishing lower bounds for their sizes in modular arithmetic over prime fields and revisiting classical results in the integers.

Contribution

It generalizes known results from integers to prime modular groups, providing new bounds and proofs for sumset structures and sizes.

Findings

Lower bounds for |h^{(r)}A| in prime modular groups

Structural characterization of sumsets with minimal size

New proofs for sumset problems in integers

Abstract

Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If $G=\mathbb{Z}$ lower bounds for $|h^{(r)}A|$ are known, as well as the structure of the sets of integers for which $|h^{(r)}A|$ is minimal. In this paper we generalize this result by giving a lower bound for $|h^{(r)}A|$ when $G=\mathbb{Z}/p\mathbb{Z}$ for a prime $p$, and show new proofs for the direct and inverse problems in $\mathbb{Z}$.