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A counterexample to symmetry of $L^p$ norms of eigenfunctions

We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $ψ_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.