Abstract:

Let $h,k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $hA:=\{a_1+a_2+\cdots+a_r:a_i \in A, r \le h\}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$, and let $n(h,A)$ denote the largest integer $n$ such that $\{1,2,\ldots,n\} \subseteq hA$. Let $n(h,k)=\max_A\:n(h,A)$, where the maximum is taken over all sets $A$ with $k$ elements. The purpose of this note is to determine $n(h,A)$ when the elements of $A$ are in arithmetic progression. In particular, we determine the value of $n(h,2)$.