On the cardinality of general $h$-fold sumsets

TL;DR

This paper establishes the best lower bounds for the size of general $h$-fold sumsets of integer sets with bounded repetitions, and characterizes the structure of sets when these bounds are tight.

Contribution

It generalizes previous results by providing optimal bounds and structural characterizations for $h$-fold sumsets with bounded element repetitions.

Findings

Derived the best lower bound for $|h^{( extbf{r})}A|$

Characterized the structure of sets achieving minimal sumset size

Extended and unified previous results in the literature

Abstract

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\ldots,k-1$. In this paper, we give the best lower bound for $|h^{(\mathbf{r})}A|$ in terms of $\mathbf{r}$ and $h$ and determine the structure of the set $A$ when $|h^{(\mathbf{r})}A|$ is minimal. This generalizes results of Nathanson, and recent results of Mistri and Pandey and also solves a problem of Mistri and Pandey.