A set $A\subseteq\newcommand{\N}{\mathbb{N}}\N$ is said to be *Golomb* if whenever $a<b \in A$ and $a'<b' \in A$ with $(b-a) = (b' - a')$, then $a=a'$ and $b=b'$. If $A\subseteq \N$ is Golomb, we let $\newcommand{\dist}{\text{dist}}\dist(A) = \{b-a:a < b \in A\}$. 

Is there a Golomb set $A\subseteq \N$ with $\dist(A) = \N\setminus\{0\}$? If not, is there a Golomb set $A$ with $\lim\inf_{n\to\infty}\frac{1}{n+1}\big(|A\cap\{1,\ldots,n\}|\big) = 1$?