**Problem.** What is the smallest cardinality $d(n)$ of a set $A$ of integer numbers such that the difference set $A-A=\{a-b:a,b\in A\}$ contains $n$ consequtive integer numbers?

It can be shown that $(1+\sqrt{4n-3})/2\le d(n)\le \frac32p(\sqrt{n})=\frac32\sqrt{n}+O(n^{21/80})$ where $p(x)$ is the smallest prime number greater or equal to $x$.

These bounds suggest the following more precise questions:

**Question 1.** Is $d(n)\le (\sqrt{2}+o(1))\sqrt{n}$?

**Question 2.** Is $d(n)=(1+o(1))\sqrt{n}$?

**Comment.** Looking at the literature, I discovered that this question has been studied by classics: Erdos, Gal (1948), Redey, Renyi (1949), Leech (1956), Whichmann (1963), Golay (1972). More information (in the context of perfect rulers) can be found here. Wichmann proved that for every $r,s\ge 0$ there exists a set $A\subset \mathbb N\cup\{0\}$ of cardinality $n=4r+s+3$ such that $A-A=[-L,L]$ where $L=4r(r+s+2)+3(s+1)$.
This gives an affirmative answer $d(n)\le \sqrt{2n}$ to Question 1.
On the other hand, much earlier Redei and Renyi (1949) proved the lower and upper bounds $1+\frac2{3\pi}< \lim_{n\to\infty}\frac{d(2n+1)^2}{2n}=\inf_{n\in\mathbb N}\frac{d(2n+1)^2}{2n}<\frac{4}3$. These lower and upper bound were improved a bit by Leech (1956) and Golay (1972). This negatively answers my Question 2 (and completes the answer given by Lucia).