Stability of perturbed sequences as a subbasis

Let $A=\{a_1<a_2<\cdots\}$ be a set of nonnegative integers, and $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. The set $A$ is a subbasis of order $h$ if $hA$ contains an infinite arithmetic progression. Furthermore, for any set $P$ of integers, a sequence $B=\{b_1, b_2, \dots\}$ is defined as a $P$-perturbation of $A$ if $b_n-a_n\in P$ for all $n$. Let $\mathbb{Z}_0$ be the set of nonnegative integers. In this paper, we prove that: (i) for any integers $k,l$ with $0\le k<l$, every $\{k,l\}$-perturbation of $\mathbb{Z}_0$ is a subbasis of order 2; (ii) for every positive integer $k$, every $\{0,3k-1,3k\}$-perturbation of $\mathbb{Z}_0$ is a subbasis of order 4. This extends a result of John R. Burke and William A. Webb [1]. Related conjectures are also posed in the paper.