Improving bounds on $\gamma$ in $B^{\pm} \to DK^{\pm}$ and $B^{\pm,0} \to D X_s^{\pm,0}$

In view of recent experimental progress in rate and CP asymmetry measurements in $B^{\pm} \to DK^{\pm}$, we reconsider information on the weak phase $\gamma$ which can be obtained from these processes. Model-independent inequalities are proven for $\sin^2\gamma$ in terms of two ratios of partial rates for $B^{\pm,0} \to D X_s^{\pm,0}$, where $X_s$ is any multiparticle charmless state carrying strangeness $\pm 1$. Good prospects are shown to exist for using these inequalities and CP asymmetry measurements in two body and multibody decays in order to improve present bounds on $\gamma$.