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A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds. II. (English) Zbl 1270.11097

From the text: This article is concerned with estimations of \(R(t)\) from below for the remainder term in Weyl’s law for the spectral counting function of certain rational \((2\ell + 1)\)-dimensional Heisenberg manifolds. Concentrating on the case of odd \(\ell \), it continues the work done in part I [Arch. Math. 92, No. 4, 344–354 (2009; Zbl 1184.11039)] which dealt with even \(\ell \).
In the present work, it is our objective to estimate R(t) from below, in order to arrive again at a statement saying that “\(R(t)\ll t^{\ell-1/4}\) in mean-square, with an unbounded sequence of values \(t\) for which \(R(t)\) becomes exceptionally large”. In fact, we are able to find for each \(\ell>1\) an explicit function \(\omega_\ell(t)\) tending to 1, such that \(R(t) = \Omega(t^{\ell-1/4}\omega_\ell(t))\).
Theorem. For any fixed positive integer \(\ell\), let \((H_\ell\backslash \Gamma_r,g_\ell)\) be a rational \((2\ell + 1)\)-dimensional Heisenberg manifold with metric \(g_\ell\). Then the error term \(R(t)\) for the associated spectral counting function satisfies \[ \limsup_{t\to\infty}\frac{R(t)}{t^{\ell-1/4}\omega_\ell(t)}>0, \] where \[ \omega_\ell(t):=\begin{cases} (\log t)^{1/4}\quad & \text{for}\;\ell\;\text{even}, \\ (\log_2 t\log_3 t)^{1/4}\quad & \text{for}\;\ell\;\text{odd}.\end{cases} \]

MSC:

11N37 Asymptotic results on arithmetic functions
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11P21 Lattice points in specified regions

Citations:

Zbl 1184.11039
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References:

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