Problem 875

Erdős Problem #875: declared status 'open'. Formalized: no. 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 $a_{n+1}-a_n\leq n^{c}$? Prize: no. Tags: additive combinatorics.