I want to know whether number rings tend to the integers as the discriminant tends to infinity.
In detail, let $n$ be a natural number and let $C(n)$ be the set of all number fields $K$ of degree $n$. For $K\in C(n)$ let $K_{\mathbb R}=K\otimes_{\mathbb Q}\mathbb R$. The real vector space  $K_{\mathbb R}$ comes with a natural inner product and the ring of integers $\mathcal{O}_K$ is a lattice in $K_{\mathbb R}$.

Here's my question: Let $R>0$ and let $B_R$ be the open ball of radius $R$ around zero in $K_{\mathbb R}$. Is it true that there is $d>0$ such that for every $K\in C(n)$ of discriminant $d_K\ge d$ one has
$$
B_R\cap\mathcal{O}_K=B_R\cap{\mathbb Z}?
$$