Fix
α
∈
[
0
,
1
)
\alpha \in [0,1)
. Consider the random walk on the circle
S
1
S^1
which proceeds by repeatedly rotating points forward or backward, with probability
1
2
\frac 12
, by an angle
2
π
α
2\pi \alpha
. This paper analyzes the rate of convergence of this walk to the uniform distribution under “discrepancy” distance. The rate depends on the continued fraction properties of the number
ξ
=
2
α
\xi =2\alpha
. We obtain bounds for rates when
ξ
\xi
is any irrational, and a sharp rate when
ξ
\xi
is a quadratic irrational. In that case the discrepancy falls as
k
−
1
2
k^{-\frac 12}
(up to constant factors), where
k
k
is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of
ξ
\xi
which allows for tighter bounds on terms which appear in the Erdős-Turán inequality.