Erdős Problem #1194 — Growth forced by unique difference representations

Let $A=\{a_1<a_2<\cdots\}$ be such that every positive integer $n$ has a unique representation $n=a_i-a_j$ with $a_i,a_j\in A$. How slowly can the numerator $a_i=a_i^{(n)}$ grow relative to $n$?