New Lattice Point Asymptotics for Products of Upper Half-planes

Let be an irreducible lattice in PSL2(R)d (d ∈ N) and z a point in the d-fold direct product of the upper half-plane. We study the discrete set of componentwise distances D( , z) ⊂ Rd defined in (2). We prove asymptotic results on the number of γ ∈ such that dist(z, γ z) is contained in strips expanding in some directions and also in expanding hypercubes. The results improve the existing error terms, [6], and generalize the best known error term for d= 1, due to Selberg.