SOLVED
Let $A\subset\mathbb{N}$ be an additive basis of order $k$. Let $B=\{b_1<b_2<\cdots\}$ be the set of integers which are the sum of $k$ or fewer distinct $a\in A$. Is it true that $b_{n+1}-b_n=O(1)$? (Where the implied constant may depend on both $A$ and $k$.)
A problem of Burr and Erdős.
Hegyvári, Hennecart, and Plagne [HHP07] showed the answer is yes for $k=2$ (in fact with $b_{n+1}-b_n\leq 2$ for large $n$) but no for $k\geq 3$.
The proof that $b_{n+1}-b_n\leq 2$ for $k=2$ is trivial, since clearly all odd numbers in $A+A$ must be the sum of two distinct elements from $A$.
