OPEN
Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite set such that the sets
\[S_r = \{ a_1+\cdots +a_r : a_1<\cdots<a_r\in A\}\]
are disjoint for distinct $r\geq 1$. How fast can such a sequence grow? How small can $a_{n+1}-a_n$ be? In particular, for which $c$ is it possible that $a_{n+1}-a_n\leq n^{c}$?
A problem of Deshouillers and Erdős (an infinite version of
[874]). Such sets are sometimes called admissible. Erdős writes 'it [is not] completely trivial to find such a sequence for which $a_{n+1}/a_n\to 1$'. It is not clear from this whether Deshouillers and Erdős knew of such a sequence.
