  This paper is aimed to study the ergodic short-term behaviour of
discretizations of circle expanding maps. More precisely, we prove some
asymptotics of the distance between the $t$-th iterate of Lebesgue measure by
the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of
order $N$ by the discretization on this grid, when $t$ is fixed and the order
$N$ goes to infinity. This is done under some explicit genericity hypotheses on
the dynamics, and the distance between measures is measured by the mean of
\emph{Cram\'er} distance. The proof is based on a study of the corresponding
linearized problem, where the problem is translated into terms of
equirepartition on tori of dimension exponential in $t$.
  A numerical study associated to this work is presented in arXiv:2206.08000
[math.DS].
