On quantum ergodicity for higher dimensional cat maps

The uploaded paper proves the desired QUE-for-all-eigenfunctions along a density-one set of integers N in all dimensions g ≥ 1, under the natural arithmetic/dynamical assumptions (ergodicity; no ratio of distinct eigenvalues a root of unity; separable characteristic polynomial; no A-invariant rational isotropic subspace). This is stated explicitly as Theorem 1.2 and framed via the ΔA(f, N) criterion in (1.7) (see the statement and setup around (1.7) and Theorem 1.2) . The proof uses an exact Egorov property, a µ-twisted time average D(n), a 4ν-th moment bound reducing to an arithmetic counting problem Q2ν(N; n), a Chinese-remainder/tensor-product factorization of operators, prime-modulus exponential-sum bounds (Bourgain), and an “anatomy of integers” argument to select density-one sets of N (outline and key lemmas) . In contrast, the candidate solution asserts that for g ≥ 2 the statement was still open as of 2024-11-07, which conflicts with this manuscript (dated 7 Nov 2024) and its complete argument establishing (1.7) along a density-one set of N for all eigenfunctions. The candidate’s reduction via uniform bounds on twisted orbit averages S_{n,N}(δ) is informative but imposes a stronger requirement than necessary; the paper circumvents this by bounding higher moments and leveraging prime-factor savings via tensor structure. Hence the paper’s result is correct and the model’s “likely open” conclusion is incorrect.