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We use the sum-of-squares theorem from number theory to constructeigenfunctions of the Laplacian on the $d$-dimensional torus, $d \geq 2$, whichvanish to any prescribed order at some point. These functions are then appliedto provide a negative answer (in dimension $d \geq 2$) to a question in thecontext of quantitative unique continuation for spectral projectors ofSchr\"odinger operators, asked by Egidi and Veseli\'c.

Laplace-eigenfunctions on the torus with high vanishing order