Let $\alpha \in \mathbb{C}$ such that $Im(\alpha) \neq 0$, $2Re(\alpha) \in \mathbb{Z}$ and $|\alpha|^2=\alpha \overline{\alpha} \in \mathbb{Z}$. This guarantees that $(1,\alpha)$ is a $\mathbb{R}$-basis of $\mathbb{C}$ and that $A = \mathbb{Z}[\alpha] = \{ u + v\alpha : u,v \in \mathbb{Z} \}$ is a subring of $\mathbb{C}$.
Define $N(u+v\alpha)=|u+v\alpha|^2 = (u+v\alpha)(u+v\overline{\alpha})=u^2 + |\alpha|^2v^2 + 2Re(\alpha)uv \in \mathbb{N}$.
For $s \in A$, $s = s_1 + \alpha s_2$, write $\lVert s \rVert_\alpha$ for $\max{(|s_1|, |s_2|)}$. If $S \subset A$ is a subset of $A$, write $\lVert S \rVert_\alpha$ for $\{ \lVert s \rVert_\alpha : s \in S \} \subset \mathbb{N}$.
My questions are :
- Show that : $(\exists C_\alpha > 0)(\forall x \in A - \{0\})(\forall a \in A$ s.t. $a \notin (x)) : \min \lVert a + (x) \rVert_\alpha \leq C_\alpha \lVert x \rVert_\alpha$. What is the best constant $C_\alpha$ ?
($(x)$ means the ideal of $A$ generated by $x$)
- Do we have a constant $C_\alpha > 0$ s.t. $(\forall x \in A - \{0\})(\forall y \in A - \{0\})(\forall a \in A) : \min \lVert a + (x) + (y) \rVert_\alpha \leq C_\alpha (\lVert x \rVert_\alpha + \lVert y \rVert_\alpha)$. What is the best constant $C_\alpha$ ? Could we generalize to $n$ ideals $(x^{(1)}),\cdots, (x^{(n)})$ ?
I solved question 1) for at least all the $x \in A - \{0 \}$ s.t. $x=x_1 + \alpha x_2$ with $x_1 , x_2 \geq 0$, with best constant $C_\alpha = \min( \max(2+2Re(\alpha), |\alpha|^2); \max(1 + 2Re(\alpha), |\alpha|^2+1))$ in this case. In my method, I solved a certain linear system of order 2 whose matrix has determinant $N(x)$, and then took the integer parts of the solutions. I guess the method could be generalized to other $x \in A - \{ 0\}$, but I didn't make the calculations since that would be lengthy.
Does someone know a simpler method to prove 1) ? What is the best constant $C_\alpha$ ?
What about 2) ? Is it true ?
Thanks in advance.