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Rudnick and Wigman (Ann. Henri Poincar{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/mathcal{N})$, as $Etoinfty$, where $E$ is the energy and $mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/mathcal{N}^{1+alpha(d)-epsilon})$, for any $epsilon>0$ and $dgeq 4$, where $alpha(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $epsilon$) when $dgeq 5$ due to the proof of the $ell^{2}$-decoupling conjecture by Bourgain and Demeter.
Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (arithmetic random waves). We study the distrib
We consider very general random integers and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diopha
Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By w
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then trans