Arithmetical structure of sumset intersections

Abstract: The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q){q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = {h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q){q=1}^{\infty}$ such that ${1,\ldots, h_0-1} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q){q=1}^{\infty}$ such that ${1, h_0 } \subseteq \mathcal{H}(A_q)$ but ${2,3, \ldots, h_0-1} \cap \mathcal{H}(A_q) = \emptyset$.