  We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which
sharpens a result of the authors and extends it to higher dimensions. As in the
prior work, the key intermediate step is to prove a microlocal version of these
estimates, which involves a phase space decomposition of these modes which is
essentially invariant under the bicharacteristic/geodesic flow. In a companion
paper, it will be seen that these sharpened estimates yield improved $L^q(M)$
bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q <
\frac{2(d+1)}{d-1}$.
