reference request: set \$ X, Y \$ of natural numbers so that any \$ n \$ natural writes only \$ n = x + y \$

There are many couples $$X, Y$$ of infinite subsets of $$mathbb {N}: = {0,1,2 dots }$$ so that any $$n en mathbb {N}$$ write only as $$n = x + y$$, with $$x in X$$ Y $$y in Y$$. An example of such a pair is $$(X, 2X)$$ where $$X$$ is the (set of values ​​of) the sequence A000695 of Moser-de Bruijn, numbers that are the sum of different powers of $$4$$ (Decomposition is a clear thought to the binary expansions of $$n, x, and$$). More generally, one can fix $$S subset N$$, both infinite and coinfinite, and take $$X$$ Y $$Y$$ as the set of numbers whose binary expansion is supported by $$S$$, resp. in its complement (the previous example corresponds to the choice of the set of odd numbers for $$S$$). This shows that there are many similar pairs.

A more general construction is: to arrange a sequence. $$(p_i) _ {i in mathbb {N}}$$ of integers $$p_i> 1$$. Leave $$X$$ Y $$Y$$ be the set of numbers respectively $$x$$ Y $$y$$ from the way
$$x: = n_0 + p_0p_1n_2 + p_0p_1p_2p_3n_4 + p_0p_1p_2p_3p_4p_5n_6 + dots$$
$$y: = p_0n_1 + p_0p_1p_2n_3 + p_0p_1p_2p_3p_4n_5 + dots phantom {xxxxxxxxxxx}$$
where $$0 le n_k for all $$k in mathbb {N}$$. Once again, decomposition comes to the fact that any $$n en mathbb {N}$$ has a unique expansion with $$0 le n_k as
$$n = n_0 + p_0n_1 + p_0p_1n_2 + p_0p_1p_2n_3 + p_0p_1p_2p_3n_4 + dots$$

However, I suspect that there could be more examples, not in the previous way.

Question: is there such a couple? $$X, Y$$ Which one is not this way? If so, what is the most general construction?