Does positive relative density imply asymptotic additive basis behaviour?

Question

First definitions: let $A, B \ \subset \mathbb{Z_{>0}}$ and $1\in A, 1\in B$. We define the relative density of $A$ with respect to $B$ to be $$rel(A, B) = \inf_n \frac{|A \cap [1,n]|}{| B \cap [1,n]|}$$ and we say a set $B$ is an asymptotic additive basis if there exists a postive integer $h$ such that each sufficiently large positive integer can be written as a sum of $h$ (not necessarily distinct) elements of $B$.

$$\text{If $rel(A,B) > 0,$ and $B$ is an asymptotic additive basis, does it follow that $A$ is, too?}$$

I have applications in mind and I would like to know if there is literature on this as it is a natural question to ask once you work with these concepts.

Answer 1

Let $B$ be the set of squares. Let $A=\{1\}\cup \bigcup_k (2^{2^{2k}},2^{2^{2k+1}}]$. Then $|A\cap [1,2^{2^{2k}}]|= (2^{2^{2k-1}}-2^{2^{2k-2}})+(2^{2^{2k-3}}-2^{2^{2k-4}})+\ldots+1\approx \sqrt{2^{2^k}}$. These are the points up to which $A$ is sparsest, so that rel$(A,B)>0$.

Of course $B$ is an asymptotic basis of order 4. But because of the large gaps, $A$ is not an asymptotic basis of any order.