  We obtain new quantitative estimates on Weyl Law remainders under dynamical
assumptions on the geodesic flow. On a smooth compact Riemannian manifold
$(M,g)$ of dimension $n$, let $\Pi_\lambda$ denote the kernel of the spectral
projector for the Laplacian, $\mathbb{1}_{[0,\lambda^2]}(-\Delta_g)$. Assuming
only that the set of near periodic geodesics over $W\subset M$ has small
measure, we prove that as $\lambda \to \infty$ $$ \int_{W}
\Pi_\lambda(x,x)dx=(2\pi)^{-n}\text{vol}_{\mathbb{R}^n}(B)\text{vol}_g(W)\,\lambda^n+O\Big(\frac{\lambda^{n-1}}{\log
\lambda }\Big),$$ where $B$ is the unit ball. One consequence of this result is
that the improved remainder holds on all product manifolds, in particular
giving improved estimates for the eigenvalue counting function in the product
setup. Our results also include logarithmic gains on asymptotics for the
off-diagonal spectral projector $\Pi_\lambda(x,y)$ under the assumption that
the set of geodesics that pass near both $x$ and $y$ has small measure, and
quantitative improvements for Kuznecov sums under non-looping type assumptions.
The key technique used in our study of the spectral projector is that of
geodesic beams.
