  Polar codes provably achieve the capacity of a wide array of channels under
successive decoding. This assumes infinite precision arithmetic. Given the
successive nature of the decoding algorithm, one might worry about the
sensitivity of the performance to the precision of the computation.
  We show that even very coarsely quantized decoding algorithms lead to
excellent performance. More concretely, we show that under successive decoding
with an alphabet of cardinality only three, the decoder still has a threshold
and this threshold is a sizable fraction of capacity. More generally, we show
that if we are willing to transmit at a rate $\delta$ below capacity, then we
need only $c \log(1/\delta)$ bits of precision, where $c$ is a universal
constant.
