Discrete subset of $\mathbb{R}^n$

Consider $$M=\left(\begin{array}{c}1\\\pi\end{array}\right)$$

For any integer $n$, we can divide the half open unit interval $[0,1)$ into intervals $A_r:=\left[\frac rn,\frac{r+1}n\right)$ for $r=0,\ldots,n-1$.

Consider the values $[k\pi]$ (where $[\,\, ]$ denotes fractional part of), for $k=1,2,3\ldots$. These values all lie in the integer span of $1,\pi$: $$[k\pi]=k\pi-\lfloor k\pi\rfloor1.$$

Further $[k\pi]\in[0,1)$ so for each $k$ we have $[k\pi]$ in some $A_r$. Eventually we will have $[k_1\pi]$ and $[k_2\pi]$ lying in the same interval $A_r$, with $k_1\neq k_2$, by the pigeonhole principle.

Thus $[k_1\pi]-[k_2\pi]$ lies in the span of $1,\pi$ and has size less than $\frac1n$. Note that $[k_1\pi]-[k_2\pi]\neq 0$ as $\pi$ irrational implies that $0$ cannot be written as a non-trivial integer combination of $1,\pi$.

Thus elements of the span of $1,\pi$, may be arbitrarily small.