Question 141
Let $\mathbb{Z}_{\gt 0}$ be the set of all positive integers. Determine all subsets $\mathcal{S}$ of $\left\{2^{0}, 2^{1}, 2^{2}, \ldots\right\}$ for which there exists a function $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ such that
$$
\mathcal{S}=\left\{f(a+b)-f(a)-f(b) \mid a, b \in \mathbb{Z}_{\gt 0}\right\} .
$$
$$
\mathcal{S}=\left\{f(a+b)-f(a)-f(b) \mid a, b \in \mathbb{Z}_{\gt 0}\right\} .
$$