Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis for $L^2(\Omega) $
Let $\Lambda \subset \mathbb{R}^n$ be a countable subset. What conditions on $\Lambda $ make it so that there exists $\Omega $ such that $L^2(\Omega)$ has $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ as an orthogonal basis?
I am curious what types of conditions we could look at. The only thing I know people have looked at is with regard to the first problem and revolves around the Fuglede Conjecture which conjectures that $\Omega $ must tiles $\mathbb{R}^n $. Is there anything other kind of geometric or algebraic condition that has been looked at?
