A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds

This article provides an Omega-result for the remainder term in Weyl’s law for the spectral counting function of certain rational (2l+ 1)-dimensional Heisenberg manifolds. Introduction. For M a closed n -dimensional Riemannian manifold with a metric g and Laplace-Beltrami operator ∆, let N(t) denote the spectral counting function N(t) := ∑ λ eigenvalue of ∆ λ≤t d(λ) where d(λ) is the dimension of the eigenspace corresponding to λ , and t is a large real variable. Then a deep and very general theorem due to L. Hörmander [6] tells us that (1.1) N(t) = vol(M) (4π)n/2 Γ( 1 2n+ 1) t +O ( t ) (”Weyl’s law”), and that the error term in general cannot be improved. Nevertheless, it is of interest to study the order of magnitude and the asymptotic behavior of the remainder R(t) = N(t)− vol(M) (4π)n/2 Γ( 12n+1) t for special manifolds M . The most classic example, namely the case that M = R/Z , the n -dimensional torus, is (equivalent to) a central problem in the theory of lattice points in large domains, namely to provide asymptotic results for the number An(x) of integer points in an origin-centered n -dimensional ball of radius x , for any dimension n ≥ 2. There exists a vast literature on this particular subject: We only refer to the works of Huxley [7], [8], Hafner [4], and Soundararajan [17] for the planar case, for the papers by Chamizo & Iwaniec [1], HeathBrown [5], and Tsang [18] for dimension n = 3, and to the monographs of Walfisz [20], and Krätzel [14], [15], as well as to the recent, quite comprehensive, survey article [9]. In fact, for M = R/Z , we see that (1) {u 7→ e(m · u) : m ∈ Z} is a basis for the eigenfunctions of the Laplace operator ∆ = − ∑n j=1 ∂jj , acting on functions from R /Z Mathematics Subject Classification (2000): 11N37, 35P20, 58J50, 11P21. (∗) The author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P20847-N18. (1) Bold face letters will denote throughout elements of R , resp., Z , which may be viewed also as (1× n) -matrices (”row vectors”) where applicable. Further, | · | stands for the Euclidean norm. into C . The corresponding eigenvalues are 4π2|m|2 , m ∈ Z . For any integer k ≥ 0, let as usual rn(k) denote the number of ways to write k as the sum of n squares. Then, for each k with rn(k) > 0, 4π k is an eigenvalue of ∆ whose eigenspace consists of all functions u 7→ ∑ |m|2=k c(m) e(m · u) , where c(m) are any complex coefficients. Its dimension obviously equals rn(k) , hence N(t) = ∑ k≥0: 4π2k≤t rn(k) = An ( √ t 2π )