Problem 336

Erdős Problem #336: declared status 'open'. Formalized: no. For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). Find the value of$$\lim_r \frac{h(r)}{r^2}.$$ Current best: A simple example of the order of a basis differing from the exact order is given by $A=\cup_{k\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$. They also proved that$$\frac{1}{4}\leq \lim_r \frac{h(r)}{r^2}\leq \frac{5}{4}.$$The best bounds known for the limit are$$\frac{1}{3}\leq \lim_r \frac{h(r)}{r^2}\leq \frac{1}{2},$$the lower bound originally due to Grekos [Gr88] and the upper bound to Nash [Na93]. The value of $h(4)$ is unknown, but it is known [Pl04] that $10\leq h(4)\leq 11$. Prize: no. Tags: additive basis, number theory.