Log-Scale Equidistribution of Zeros of Quantum Ergodic Eigensections

Under suitable hypotheses, a symplectic map can be quantized as a sequence of unitary operators acting on the Nth powers of a positive line bundle over a Kahler manifold. We show that if the symplectic map has sufficiently fast polynomial decay of correlations, then there exists a density one subsequence of eigensections whose masses and zeros become equidistributed in balls of logarithmically shrinking radii of lengths \(|\log N |^{-\gamma }\) for some constant \(\gamma > 0\) independent of N.