We consider the ensemble of random Gaussian Laplace eigenfunctions on T 3 = R 3 /Z 3 ('3d arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or 'energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.
Abstract. We consider the problem of finding small prime gaps in various sets C ⊂ N. Following the work of Goldston-Pintz-Yıldırım, we will consider collections of natural numbers that are wellcontrolled in arithmetic progressions. Letting qn denote the n-th prime in C, we will establish that for any small constant ǫ > 0, the set {qn|q n+1 − qn ≤ ǫ log n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that C has bounded prime gaps. Specific examples, such as the case where C is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.
We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral Quasi-Correlations, i.e. unusually small sums of lattice points lying on the same circle.
For 𝑋(𝑛) a Rademacher or Steinhaus random multiplicative function, we consider the random polynomialsand show that the 2𝑘th moments on the unit circletend to Gaussian moments in the sense of mean-square convergence, uniformly for 𝑘 ≪ (log 𝑁∕ log log 𝑁) 1∕3 , but that in contrast to the case of independent and identically distributed coefficients, this behavior does not persist for 𝑘 much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for 𝑃 𝑁 (𝜃), previously obtained in unpublished work of Harper by different methods, and(ii) show that asymptotically almost surelyfor all 𝜀 > 0.M S C 2 0 2 0 11K65 (primary), 11N37, 42A05 (primary)
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