New lattice point asymptotics on products of upper half-planes

BRUGGEMAN, R. W., GRUNEWALD F., MIATELLO R.J.,

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2011 vol. 2011 p. 1510 - 1519

1073-7928

Abstract. Given  an irreducible lattice in PSL2(R)^d , (d ge  2) and z a point in the d-fold direct product of the upper half plane, we study the discrete set of componentwise distances D(gamma; z) in  Rd defined in (1). We prove asymptotic results on the number of 2  such that d(z; z) is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes improve the existing error terms  by Gorodnik-Nevo [6],  and generalize the Selberg error term for d = 1.bstract. Given  an irreducible lattice in PSL2(R)^d , (d ge  2) and z a point in the d-fold direct product of the upper half plane, we study the discrete set of componentwise distances D(gamma; z) in  Rd defined in (1). We prove asymptotic results on the number of 2  such that d(z; z) is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes improve the existing error terms  by Gorodnik-Nevo [6],  and generalize the Selberg error term for d = 1.