The circle problem in the hyperbolic plane. (English) Zbl 0812.11035

The hyperbolic circle problem asks for good asymptotic estimates for the number of points of an orbit of a cofinite Fuchsian group lying in a large circle. It is possible to attack these problems using the spectral theory of Fuchsian groups. In this paper the authors consider the “error term”, i.e. the difference between the expected and actual number of points. The authors use Selberg’s theory to determine the asymptotic behaviour of the averages of this quantity. Moreover, they can use similar methods (which have to be augmented with a remarkably simple result concerning diophantine approximation) to obtain a variety of \(\Omega\)-results, which show that the influence of the spectrum with \(\text{Re} (s)=1/2\) is essentially as strong as one would expect. They extend the asymptotic estimate to cofinite discrete groups acting on hyperbolic spaces of arbitrary dimension. They also describe the results of numerical experiments.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11N75 Applications of automorphic functions and forms to multiplicative problems
11P21 Lattice points in specified regions
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
Full Text: DOI