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 $N$th powers of a positive line bundle over a K\"{a}hler manifold. We show that if the symplectic map has 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 $\lvert \log N \rvert^{-\gamma}$ for some constant $\gamma > 0$ independent of $N$.