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 $$\liminf_{n\to\infty}\frac{|\dist(A)\cap\{1,\ldots,n\}|}{n} = 1?$$
Yes, with a straightforward induction.
Let $F$ be a finite Golomb subset. Let $n\ge 1$ be the smallest element not in $\mathrm{dist}(F)$.
Define $F_m=F\cup\{m,m+n\}$. Define $u(F)=F_m$ where $m$ is the smallest number such that $F_m$ is also Golomb. To prove this is well-defined, it is enough to show that there exists $m$ such that $F_m$ is Golomb, and actually that every large enough $m$ works. Indeed, suppose that $m$ is large enough. Take two distinct increasing pairs and show they have distinct distances. If both are in $F$, then it's OK. If one is in $F$ and the other is $(m,m+n)$, it's OK. If one is in $F$ and one is half in $F$, it's OK (provided $m> 2\max(F)$). If both are half in $F$, say $(f_1,m)$, $(f_2,m+n)$, we have $m-f_1=m+n-f_2$, i.e., $f_2-f_1=n$, contradiction.
Hence define $X_0=\emptyset$, $X_{i+1}=u(X_i)$. Then $X=\bigcup_i X_i$ works.
For instance $$X_{15}=\{1,2,5,7,15,22,38,47,65,76,120,132,154,173,$$ $$241,265,327,353,482,510,575,605,786,821,984,$$ $$1023,1151,1198,1382,1430\}.$$ (It's not in OEIS, apparently.)
