The remainder in Weyl’s law for 𝑛-dimensional Heisenberg manifolds

Abstract

We prove that the error term in Weyl’s law for ‘rational’ ( 2 n + 1 ) (2n+1) -dimensional Heisenberg manifolds is of order O ( t n − 7 / 41 ) O(t^{n-{7}/{41}}) . In the ‘irrational’ case, for generic ( 2 n + 1 ) (2n+1) -dimensional Heisenberg manifolds with n > 1 n>1 , we prove that the error term is of the order O ( t n − 1 / 4 log ⁡ t ) O(t^{n-{1}/{4}}\log t) . The polynomial growth is optimal.