This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace
eigenfunctions
${\varphi}_{j}$
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,
$${N}_{\mathit{\epsilon},H}^{c}(\lambda ):=\sum _{j,{\lambda}_{j}\le \lambda}\sum _{k:{\mu}_{k}c{\lambda}_{j}<\mathit{\epsilon}}{\int}_{H}{\varphi}_{j}{\overline{\psi}}_{k}d{V}_{H}{}^{2},$$ 
where
$\{{\lambda}_{j}\}$ are the
eigenvalues of
${\sqrt{\mathrm{\Delta}}}_{M}$,
and
$\{({\mu}_{k},{\psi}_{k})\}$
are the eigenvalues and the corresponding eigenfunctions of
${\sqrt{\mathrm{\Delta}}}_{H}$. The inner sums represent
the “jumps” of
${N}_{\mathit{\epsilon},H}^{c}(\lambda )$ and reflect the
geometry of geodesic
$c$biangles
with one leg on
$H$
and a second leg on
$M$
with the same endpoints and compatible initial tangent vectors
$\xi \in {S}_{H}^{c}M$,
${\pi}_{H}\xi \in {B}^{\ast}H$, where
${\pi}_{H}\xi $ is the orthogonal
projection of
$\xi $
to
$H$. A
$c$biangle occurs
when
$\left{\pi}_{H}\xi \right\u2215\left\xi \right=c$. Smoothed
sums in
${\mu}_{k}$ are
also studied and give sharp estimates on the jumps. The jumps themselves may jump as
$\mathit{\epsilon}$ varies, at certain
values of
$\mathit{\epsilon}$ related to
periodicities in the
$c$biangle
geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps.
The results refine those of our previous article, where the inner sums run over
$k$ such
that
${\mu}_{k}\u2215{\lambda}_{j}c\le \mathit{\epsilon}$
and where geodesic biangles do not play a role.
