  A general parametrization of the $B^+\to\pi^+ K^0$ and $B_d^0\to\pi^- K^+$
decay amplitudes is presented. It relies only on the isospin symmetry of strong
interactions and the phase structure of the Standard Model and involves no
approximations. In particular, this parametrization takes into account both
rescattering and electroweak penguin effects, which limit the theoretical
accuracy of bounds on $\gamma$ arising from the combined $B^\pm\to\pi^\pm K$,
$B_d\to\pi^\mp K^\pm$ branching ratios. Generalized bounds making also use of
the CP asymmetry in the latter decay are derived, and their sensitivity to
possible rescattering and electroweak penguin effects is investigated. It is
pointed out that experimental data on $B^\pm\to K^\pm K$ allow us to include
rescattering processes in these bounds completely, and an improved theoretical
treatment of electroweak penguins is presented. It is argued that rescattering
effects may enhance the combined $B^\pm\to K^\pm K$ branching ratio by a factor
of ${\cal O}(10)$ to the $10^{-5}$ level, and that they may be responsible for
the small present central value of the ratio of the combined $B_d\to\pi^\mp
K^\pm$ and $B^\pm\to\pi^\pm K$ branching ratios, which has recently been
reported by the CLEO collaboration and, if confirmed, would exclude values of
$\gamma$ within a large region around $90^\circ$.
