# Shattering with sinusoids

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.

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