Sumsets with a minimum number of distinct terms
Jagannath Bhanja
TL;DR
This work introduces the generalized sumset $h^{(≥r)}A$, unifying $hA$ and the restricted sumset, and derives an explicit upper bound on the minimal size of $h^{(≥r)}A$ over $\mathbb{Z}_m$ via a constructive family $A_d(m,k)$. It also yields a sharp lower bound in $\mathbb{Z}_p$ and over $\mathbb{Z}$, with arithmetic progression structure arising when bounds are tight, connecting to classical results such as Cauchy–Davenport, Dias da Silva–Hamidoune, and Lev-type bounds. The analysis relies on decomposing $h^{(≥r)}A$ into a sum of a lower-order and a restricted-sum component and explores direct/inverse consequences, including potential Freiman-type structural results, and discusses the dual sumset $h^{(≤r)}A$ and related open questions. Overall, the paper advances understanding of sumsets with a minimum number of distinct terms and lays groundwork for further inverse and structural results in additive combinatorics.
Abstract
For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set by $h^{(\geq r)}A$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. As the main result of this article, we give an upper bound for the minimum size of $h^{(\geq r)}A$ over $\mathbb{Z}_m$ for $m \geq 2$. Further, by an observation relating the sumsets $hA$, $h\hat{}A$, and $h^{(\geq r)}A$ we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ and also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number.