Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more information on $M$. For example, if $n=1$, it is easy to see that $M$ has to be a subgroup of $\mathbb{R}$, thanks to the classical characterization of such subgroups. Is it still true when $n\geq2$?



Closed cobounded additive submonoid of $\mathbb{R}^n$
Yes, it has to be a subgroup. Fix $v\in M$. We need to prove that $-v\in M$. It is sufficient to find an element of $M$ arbitrarily close to $-v$.


Choose $u\_1,\ldots,u\_n\in \mathbb{R}^n$ so that $v,u\_1,\ldots,u\_n$ are the vertices of a regular simplex with center at the origin. Choose large $N$ and consider the points $w\_i\in M$ such that $\|Nu\_i-w\_i\|\leqslant r$. We may choose $N$ so large that the coordinates of $v$ in the basis $\{w\_1,\ldots,w\_n\}$ are negative: $v=-\sum\_{i=1}^n t\_i w\_i,0<t\_i$ (this is because $w\_i/N$ is close to $u\_i$ for all $i$, and the coordinates of $v$ in the basis $\{u\_1,\ldots,u\_n\}$ are all equal to -1.) 


Now by Kronecker approximation theorem we may find a positive integer $s$ so that $st\_1,\ldots,st\_n$ are almost integers, denote $st\_i=k\_i+\varepsilon\_i$, where $k\_i$ are non-negative integers and $\varepsilon\_i$ are small. Then $$M\ni (s-1)v+\sum k\_iw\_i=-v-\sum \varepsilon\_i w\_i$$ 
is close to $v$. 


The closed subgroups of $\mathbb{R}^n$ are direct sums of subspaces and lattices. And the non-empty ball condition reads simply as being full-rank. 


