  Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary
steps are lattice variables with span $\alpha \not\in \mathbb{Q}$ or $p/q \in
\mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational
case we have a random walk on the finite cyclic subgroup $\mathbb{Z}_q$, and
the central limit theorem and the law of the iterated logarithm follow from
classical results on finite state space Markov chains. In this paper we extend
these results to random walks with irrational span $\alpha$, and explicitly
describe the transition of these Markov chains from finite to general state
space as $p/q \to \alpha$ along the sequence of best rational approximations.
We also consider the rate of weak convergence to the stationary distribution in
the Kolmogorov metric, and in the rational case observe a surprising transition
from polynomial to exponential decay after $\approx q^2$ steps; this seems to
be a new phenomenon in the theory of random walks on compact groups. In
contrast, the rate of weak convergence to the stationary distribution in the
total variation metric is purely exponential.
