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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.

Inspired by this question

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) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\limits_{i=1}^K z_i = 0$, 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.

Inspired by this question

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

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.

Inspired by this question