It is well known that for a set $A$ of integers, if $gcd(A) = 1$, then the (integer) linear combination of $A$ is $\mathbb{Z}$. I'm looking for the probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in \mathbb{N}^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)/N|\geq \varepsilon$; and $N/\sum_n\alpha(n)\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have: $|CCS(\alpha)|/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.