Geometric condition for imaginary quadratic integer ring to be norm-Euclidean

Statement

Suppose ${\mathcal {O}}$ is an imaginary quadratic integer ring. Then, ${\mathcal {O}}$ is a norm-Euclidean ring of integers if and only if the following is true: for any point $z\in \mathbb {C}$, there exists a point $w\in {\mathcal {O}}$ such that:

$\left|z-w\right|<1$.

Related facts

Applications

• Classification of norm-Euclidean imaginary quadratic integer rings

Facts used

1. The norm in an imaginary quadratic integer ring is simply the square of the modulus as a complex number.
2. The field of fractions of ${\mathcal {O}}$ is dense in $\mathbb {C}$.