# Closed cobounded additive submonoid of $\mathbb{R}^n$

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" in the topological sense or is it a terminology about monoids?
– efs
Commented Jan 16, 2020 at 13:19
• Yes in the topological sense! Commented Jan 16, 2020 at 16:42
• The geometric property you mentioned is known as "cobounded" (I added it in the title, you can revert if you like)
– YCor
Commented Mar 11, 2020 at 9:23

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: $$v=-\sum_{i=1}^n t_i w_i,0 (this is because $$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.