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 <p_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 <p_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?