Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let $\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals. Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{B} \cap \overline{\mathbb{Q}}$.

**Questions:**

Is $\mathcal{B}_{\overline{\mathbb{Q}}}$ dense in $\mathcal{B}$ in the sense that for every $z \in \mathcal{B}$ and every $\varepsilon > 0$, the $\varepsilon$-neighborhood of $z$ has nontrivial intersection with $\mathcal{B}_{\overline{\mathbb{Q}}}$?

Given $z \in \mathbb{C}$, let $\mathcal{B} + z := \{b + z \ | \ b \in \mathcal{B}\}$ denote the translate of $\mathcal{B}$ by $z$. Is there a $z \in \mathbb{C}$ such that $(\mathcal{B} + z) \cap \overline{\mathbb{Q}} = \emptyset$?

What are the answers to (1.) and (2.) if we replace $\overline{\mathbb{Q}}$ by the Gaussian rationals $\mathbb{Q}[i]$ or by the cyclotomic integers $\mathbb{Z}[e^{\frac{2\pi i}{5}}]$?

integerswhose Galois orbits accumulate to $\mathcal{B}$. What you ask, with the algebraic numbers lying exactly on $\mathcal{B}$, is entirely different of course. Still I thought I would make this remark. $\endgroup$1more comment