Before revision:  Furthermore, as the lattice parameter decreases, searching our data structure yields a six-approximation of the nearest diagram in \Gamma in O((m   +m^2)\log\tau) time and a constant factor approximation of the kth nearest diagram in O((m   +m^2+k)\log\tau) time.
After revision:  Furthermore, as the lattice parameter decreases, searching our data structure yields a six-approximation of the nearest diagram in \Gamma in O((m  +m^2)\log\tau) time and a constant factor approximation of the kth nearest diagram in O((m  \log{n +m^2+k)\log\tau) time.