In this work we study a modification of the hyperbolic circle problem, which is
one of the problems originally studied by A. Good. We consider the orbit of double
cosets of a Fuchsian group Γ by two hyperbolic subgroups H_1, H_2 in the hyperbolic
plane.
We use a relative trace formula with suitable test functions for the counting of
lengths of geodesic segments perpendicular to the closed geodesics corresponding
to H_1 and H_2. We present an elementary proof providing the main term in the
asymptotics and an error termof order O(X^{2/3}). We study themean square of the
error termand prove that it is consistent with the conjectural optimal error term
O(X^{1/2+ϵ}). To apply the relative trace formula we develop a large sieve inequality
for periods ofMaass forms. This requires a more subtle understanding of Huber’s
transform, which is a special case of the Jacobi transform studied by Flensted-
Jensen and Koornwinder. Our counting problem is a special case of counting in
the orthospectrum. We are motivated by previous work on geodesic segments
between a point and a closed geodesic, studied by Huber and Chatzakos–Petridis.