  This article concerns joint asymptotics of Fourier coefficients of
restrictions of Laplace eigenfunctions $\phi_j$ of a compact Riemannian
manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and
study the asymptotics of the thin sums, $$ N^{c} _{\epsilon, H }(\lambda): =
\sum_{j, \lambda_j \leq \lambda} \sum_{k: |\mu_k - c \lambda_j | < \epsilon}
\left| \int_{H} \phi_j \overline{\psi_k}dV_H \right|^2 $$ where $\{\lambda_j\}$
are the eigenvalues of $\sqrt{-\Delta}_M,$ and $\{(\mu_k, \psi_k)\}$ are the
eigenvalues, resp. eigenfunctions, of $\sqrt{-\Delta}_H$. The inner sums
represent the `jumps' of $ N^{c} _{\epsilon, H }(\lambda)$ and reflect the
geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$
with the same endpoints and compatible initial tangent vectors $\xi \in S^c_H
M, \pi_H \xi \in B^* H$, where $\pi_H \xi$ is the orthogonal projection of
$\xi$ to $H$. A c-bi-angle occurs when $\frac{|\pi_H \xi|}{|\xi|} = c$.
Smoothed sums in $\mu_k$ are also studied, and give sharp estimates on the
jumps. The jumps themselves may jump as $\epsilon$ varies, at certain values of
$\epsilon$ related to periodicities in the c-bi-angle geometry. Subspheres of
spheres and certain subtori of tori illustrate these jumps. The results refine
those of the previous article (arXiv:2011.11571) where the inner sums run over
$k: | \frac{\mu_k}{\lambda_j} - c| \leq \epsilon$ and where geodesic bi-angles
do not play a role.
