The maximum size of sumsets in finite cyclic groups

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^ A is the set of all sums of h distinct elements of A. The union of restricted sumsets s^ A, where s=0, 1, …, h, is denoted by [0, h]^ A. For fixed positive integers m and h, the maximum size of the sumset $hA$ of a set A ⊆ G with m elements is denoted by $ν(G, m, h)$. In other words, ν(G, m, h) = max\|hA| : A ⊆ G, |A|= m\. Analogous quantities can be defined for the sumsets h^ A and [0, h]^ 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^ A and [0, h]^ A. We prove that, for any positive integer h, there are infinitely many positive integers m and n such that ν(Zₙ, m, h) is strictly less than the optimal upper bound. We also prove similar results for quantities related to the sumsets h^ A and [0, h]^ A also. These results provide the partial solutions to the problems posed by Bajnok.