I am interested in characterizing sets for which the number of different subset-sums is bounded by a polynomial in the size of the set. Formally:
Let $A = \langle a_1, a_2, \dots, a_m \rangle$ be a multi-set of non-negative integers (repetitions possible). For a subset $A'\subseteq A$, let $s(A')$ denote the sum of elements in $S'$. Then the number of different subset-sums for $A$ is given by $\chi(A) = |\{s(A') : A'\subseteq A\}|$.
In general, $\chi(A)$ can be as large as $2^m$, for $m = |A|$, e.g. when $A = \langle1,2,\dots,2^{m-1}\rangle$. On the other hand, when each $a_i \in [k]$, then the sum of any subset of $A$ lies in $\{0,1,\dots,k\cdot m\}$, and hence $\chi(A) \le k\cdot m+1$, which is polynomial in $m$ if $k = poly(m)$.
I would like to know if there is a general characterization of sets $A$ for which $\chi(A)\le poly(|A|)$? If this is hard, are there other large classes of sets $A$ for which $\chi(A)$ is reasonably small?