The maximum size of sumsets in finite cyclic groups

Abstract

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. The union of restricted sumsets $s^\wedge A$, where $s=0, 1, \ldots, h$, is denoted by $[0, h]^\wedge A$. For fixed positive integers $m$ and $h$, the maximum size of the sumset $hA$ of a set $A \subseteq G$ with $m$ elements is denoted by $\nu(G, m, h)$. In other words, $\nu(G, m, h) = \max\{|hA| : A \subseteq G, |A|= m\}$. Analogous quantities can be defined for the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. Optimal upper bounds are known for these quantities. If $G$ is a finite cyclic group of order $n$, then each of these quantities agrees with the optimal upper bound, except in many cases. Bajnok posed the problem of determining all positive integers $n$, $m$, and $h$ for which the value of the function $f(n, m, h)$ is strictly less than the optimal upper bound. He posed similar problems for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. We prove that, for any positive integer $h$, there are infinitely many positive integers $m$ and $n$ such that $\nu(\mathbb{Z}_n, m, h)$ is strictly less than the optimal upper bound. We also prove similar results for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$ also. These results provide the partial solutions to the problems posed by Bajnok.