  We consider the problem of multiple-input single-output Broadcast Channels
with Rayleigh fading where the transmitter has access to delayed knowledge of
the channel state information. We first characterize the capacity region of
this channel with two users to within constant number of bits for all values of
the transmit power. The proposed signaling strategy utilizes the delayed
knowledge of the channel state information and the previously transmitted
signals, in order to create a signal of common interest for both receivers.
This signal would be the quantized version of the summation of the previously
transmitted signals. A challenge that arises in deriving the result for finite
signal-to-noise ratio regimes is the correlation that exists between the
quantization noise and the signal. To guarantee the independence of
quantization noise and signal, we extend the framework of lattice quantizers
with dither together with an interleaving step. For converse, we use the fact
that the capacity region of this problem is upper-bounded by the capacity
region of a physically degraded broadcast channel with no channel state
information where one receiver has two antennas. Then, we derive an outer-bound
on the capacity region of this degraded broadcast channel. Finally, we show how
to extend our results to obtain the approximate capacity of the $K$-user
multiple-input single-output Broadcast Channel with delayed knowledge of the
channel state information at the transmitter to within $2 \log_2 \left( K + 2
\right)$ bits/s/Hz.
