Abstract
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
/
𝒩
1
+
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
ℓ
2
{\ell^{2}}
-decoupling conjecture by
Bourgain and Demeter.
On the variance of the nodal volume of arithmetic random waves
