Question 4
Let $\mathbb{Z}_{ \gt 0}$ be the set of all positive integers. Determine all subsets $\mathcal{S}$ of $\{2^0, 2^1, 2^2, \ldots\}$ for which there exists a function $f: \mathbb{Z}_{ \gt 0} \to \mathbb{Z}_{ \gt 0}$ such that
$$ \mathcal{S}=\{f(a+b)-f(a)-f(b)\mid a,b\in\mathbb{Z}_{ \gt 0}\}. $$
$$ \mathcal{S}=\{f(a+b)-f(a)-f(b)\mid a,b\in\mathbb{Z}_{ \gt 0}\}. $$