Random waves on $\mathbb{T}^3$: nodal area variance and lattice point correlations

On the variance of the nodal volume of arithmetic random waves

“…We stress that the results in [11] and [2] are stronger, because they obtain true asymptotics for the variance Var(V). In two and three dimensions the limiting distribution of V has also been determined, see [14] and [7].…”

“…Remark 1. In Proposition 1.2 we show explicitly the dependence on |X (4)| so as to preserve the analogy with [11] and [2]. However, for the purpose of proving Theorem 1.1, the distinction between |X (4)| and |C(4)| is superfluous since we merely bound the former by the latter.…”

“…In this section we prove Proposition 1.2 under the assumption of two technical results that will be established in sections 4 and 5. Our argument follows a method successfully employed in [11] and [2], which itself builds on ideas from [15] and [19]. The starting point is a detailed analysis of the Kac-Rice formulas, which allows us to express the variance Var(V) in terms of the covariance function…”

“…was later extended to d = 3 by Benatar and Maffucci [2] who proved an asymptotic of the form c 3 E/N 2 with c 3 = (2/375)G 2 3 . Notice that the leading term is again smaller than expected due to the same cancellation phenomenon.…”

“…In two and three dimensions the limiting distribution of V has also been determined, see [14] and [7]. On the other hand, when d ≥ 4, by a straightforward analytic argument (see [2, p. 3050]), it is possible to show the lower bound 2) , for any ℓ ≥ 4. In light of Proposition 1.2 and the fact that |C(4)| and |X (4)| are roughly of the same order (see section 2.1), (1.7) means that the upper bound in Theorem 1.1 is the best possible with the current method for all d ≥ 5 up to the factor N ǫ .…”