Rudnick and Wigman (2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d -dimensional torus is O(E/𝒩){O(E/\mathcal{N})}, as E→∞{E\to\infty}, where E is the energy and 𝒩{\mathcal{N}} is the dimension of the eigenspace corresponding to E . Previous results have established this with stronger asymptotics when d=2{d=2} and d=3{d=3}. In this brief note we prove an upper bound of the form O(E/𝒩1+α(d)-ϵ){O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})}, for any ϵ>0{\epsilon>0} and d≥4{d\geq 4}, where α(d){\alpha(d)} is positive and tends to zero with d . The power saving is the best possible with the current method (up to ϵ) when d≥5{d\geq 5} due to the proof of the ℓ2{\ell^{2}}-decoupling conjecture by Bourgain and Demeter.
In this paper, we introduce a new technique in the study of the *{*}-regular closure of some specific group algebras KG inside 𝒰(G){{\mathcal{U}}(G)}, the *{*}-algebra of unbounded operators affiliated to the group von Neumann algebra 𝒩(G){{\mathcal{N}}(G)}. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form CK(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, where X is a totally disconnected compact metrizable space, T is a homeomorphism of X , and CK(X){C_{K}(X)} stands for the algebra of locally constant functions on X with values on an arbitrary field K . The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of ℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.
In this paper, we introduce a new technique in the study of the *{*}-regular closure of some specific group algebras KG inside 𝒰(G){{\mathcal{U}}(G)}, the *{*}-algebra of unbounded operators affiliated to the group von Neumann algebra 𝒩(G){{\mathcal{N}}(G)}. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form CK(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, where X is a totally disconnected compact metrizable space, T is a homeomorphism of X , and CK(X){C_{K}(X)} stands for the algebra of locally constant functions on X with values on an arbitrary field K . The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of ℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.
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