  In this series, we investigate quantum ergodicity at small scales for linear
hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove
quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck
constant. First, for all integers $N\in\mathbb{N}$, we show quantum ergodicity
at logarithmical scales $|\log h|^{-\alpha}$ for some $\alpha>0$. Second, we
show quantum ergodicity at polynomial scales $h^\alpha$ for some $\alpha>0$, in
two special cases: $N\in S(\mathbb{N})$ of a full density subset
$S(\mathbb{N})$ of integers and Hecke eigenbasis for all integers.
