Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an eigenfunction of the Laplacian $f: \mathbb{T}^d \to \mathbb{R}$ such that $sgn(f(\theta_i)) = z_i$ for all $i$?

Here sgn is signum function.

Motivated by this [question][1]


  [1]: https://mathoverflow.net/q/307363/14414