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Even a much weaker consequence, that no eigenfunction has unusually high peaks, is unproven.","posed_since":"1977","precise":"Compact negatively curved surface $M$ (or an ergodic billiard), $-\\operatorname{Lap} \\phi_{j} = \\lambda_{j} \\phi_{j}$, $||\\phi_{j}||_{2} = 1$. Prove that for a density-one subsequence the distribution of $\\phi_{j}(x)$, $x$ uniform on $M$, converges to the Gaussian of variance $1/\\operatorname{vol}(M)$; a weaker target is the sup-norm bound ||phi_j||_inf <= C_eps lambda_j^eps. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of Gaussian moments (e.g. ||phi_j||_4^4 -> 3/vol(M)) for a density-one subsequence on one negatively curved surface.","status_note":"The general sup-norm bound is $\\lambda^{1/4}$ up to logarithmic improvement; on arithmetic surfaces Iwaniec and Sarnak (1995) proved $\\lambda^{5/24+\\varepsilon}$.","title":"Gaussian value distribution of chaotic 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