OPEN
This is open, and cannot be resolved with a finite computation.
Let $F(N)$ be the smallest possible size of $A\subset \{0,1,\ldots,N\}$ such that $\{0,1,\ldots,N\}\subset A-A$. Find the value of\[\lim_{N\to \infty}\frac{F(N)}{N^{1/2}}.\]
The
Sparse Ruler problem. Rédei asked whether this limit exists, which was proved by Erdős and Gál
[ErGa48]. Bounds on the limit were improved by Leech
[Le56]. The limit is known to be in the interval $[1.56,\sqrt{3}]$. The lower bound is due to Leech
[Le56], the upper bound is due to Wichmann
[Wi63]. Computational evidence by Pegg
[Pe20] suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\subset \{0,1,\ldots,N\}$.
