  Gap Hamming Distance is a well-studied problem in communication complexity,
in which Alice and Bob have to decide whether the Hamming distance between
their respective n-bit inputs is less than n/2-sqrt(n) or greater than
n/2+sqrt(n). We show that every k-round bounded-error communication protocol
for this problem sends a message of at least Omega(n/(k^2\log k)) bits. This
lower bound has an exponentially better dependence on the number of rounds than
the previous best bound, due to Brody and Chakrabarti. Our communication lower
bound implies strong space lower bounds on algorithms for a number of data
stream computations, such as approximating the number of distinct elements in a
stream.
  Subsequent to this result, the bound has been improved by some of us to the
optimal Omega(n), independent of k, by using different techniques.
