Manifolds whose Weyl spectral asymptotics have small but not tiny remainders

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A compact, n -dimensional Riemannian manifold M has Weyl spectral asymptotics with remainder E_{M}(R) ; i.e., the spectral counting function satisfies \mathcal{N}(\Delta_{M},R)=C(M)R^{n}+ E_{M}(R) , with E_{M}(R)=o(R^{n}) . Generally, one actually has E_{M}(R)=O(R^{n-1}) , and one seeks geometrical conditions under which stronger estimates hold on the remainder and also conditions limiting how extra small the remainder can be. Here, we produce n -dimensional manifolds whose Weyl remainders are o(R^{n-1}) but not O(R^{n-1-\alpha}) for any \alpha>0 .