1.1 Shnirelman’s theorem and small-scale equidistribution

Assuming without loss of generality that ${\mathcal{M}}$ is unit volume $\operatorname{Vol}({\mathcal{M}})=1$ , the celebrated Shnirelman’s theorem [Reference Colin de Verdière9, Reference Shnirelman27, Reference Zelditch29] asserts that if ${\mathcal{M}}$ is chaotic (i.e., the geodesic flow on ${\mathcal{M}}$ is ergodic), then “most” of the $\{\unicode[STIX]{x1D719}_{j}\}$ are $L^{2}$ -equidistributed. In particular, they are equidistributed in position space, i.e., there exists a density- $1$ sequence $j_{k}$ such that for all “nice” domains ${\mathcal{A}}\subseteq {\mathcal{M}}$ , we have

(1.1) $$\begin{eqnarray}\displaystyle \lim _{k\rightarrow \infty }\int _{{\mathcal{A}}}\unicode[STIX]{x1D719}_{j_{k}}(x)^{2}\,dx=\operatorname{Vol}({\mathcal{A}}). & & \displaystyle\end{eqnarray}$$

Beyond Shnirelman’s theorem, Berry’s universality conjecture [Reference Berry3, Reference Berry4] implies that for a generic chaotic manifold (1.1) holds for ${\mathcal{A}}$ shrinking with $k$ slower than the Planck scale $E_{\!j_{k}}^{-1/2}$ . More precisely, it states that there exists a density- $1$ sequence $\{{j_{k}\}}_{k}$ so that if $r_{0}(E):\mathbb{R}_{{>}0}\rightarrow \mathbb{R}_{{>}0}$ satisfies $r_{0}(E)\cdot E^{1/2}\rightarrow \infty$ diverging arbitrarily slowly, then, for $B_{x}(r)$ the radius- $r$ geodesic ball in ${\mathcal{M}}$ centred at $x$ , we have

(1.2) $$\begin{eqnarray}\displaystyle \biggl|\int _{B_{x}(r)}\unicode[STIX]{x1D719}_{j_{k}}(y)^{2}\,dy-\operatorname{Vol}(B_{x}(r))\biggr|=o_{k\rightarrow \infty }(r^{d}) & & \displaystyle\end{eqnarray}$$

uniformly for all $x\in {\mathcal{M}}$ and $r>r_{0}(E_{\!j_{k}})$ , i.e.,

(1.3) $$\begin{eqnarray}\displaystyle \sup _{\substack{ x\in {\mathcal{M}} \\ r>r_{0}(E_{\!j_{k}\!})}}\biggl|\frac{\int _{B_{x}(r)}\unicode[STIX]{x1D719}_{j_{k}}(y)^{2}\,dy}{\operatorname{Vol}(B_{x}(r))}-1\biggr|\rightarrow 0. & & \displaystyle\end{eqnarray}$$

The following recent results are rigorous manifestations of the small-scale (“shrinking balls”) statement (1.3). Luo and Sarnak [Reference Luo and Sarnak24, Theorem 1.2] established the small-scale equidistribution for Laplace eigenfunctions on the modular surface (assuming in addition that they are Hecke eigenfunctions), where $r>E^{-\unicode[STIX]{x1D6FC}}$ with a small $\unicode[STIX]{x1D6FC}>0$ , and Young [Reference Young28], conditionally on the generalized Riemann hypothesis, refined this estimate for $r>E^{-1/6+o(1)}$ holding for all such eigenfunctions. Hezari and Rivière [Reference Hezari and Rivière19] and independently Han [Reference Han15] established the equidistribution for Laplace eigenfunctions on manifolds of negative curvature on logarithmic scale (i.e., $r>(\log E)^{-\unicode[STIX]{x1D6FC}}$ for some $\unicode[STIX]{x1D6FC}>0$ ) and Han [Reference Han16] considered random Laplace eigenfunctions on “symmetric” manifolds of high spectral degeneracy; here the higher the spectral degeneracy the smaller the allowed scale is. More recently, Han and Tacy [Reference Han and Tacy17] proved the small-scale equidistribution for random Gaussian combinations of eigenfunctions on compact manifolds for $r>E^{-1/2+o(1)}$ and de Courcy-Ireland [Reference de Courcy-Ireland10] showed that, with high probability, the $L^{2}$ -mass of random Gaussian spherical harmonics is, up to a small error, equidistributed, slightly above the Planck scale.

1.2 Toral Laplace eigenfunctions

For the $d$ -dimensional torus $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ , $d\geqslant 2$ , there are high spectral degeneracies; in this case Lester and Rudnick [Reference Lester and Rudnick23, Theorem 1.1] proved that the small-scale equidistribution is satisfied by a generic Laplace eigenfunction (also considered by Hezari and Rivière [Reference Hezari and Rivière18]). More precisely, they showed that every orthonormal basis $\{\unicode[STIX]{x1D719}_{j}\}$ admits a density-1 subsequence $\{\unicode[STIX]{x1D719}_{j_{k}}\}$ of Laplace eigenfunctions obeying (1.3), with $r_{0}(E)=E^{-\unicode[STIX]{x1D6FC}(d)}$ , where $\unicode[STIX]{x1D6FC}(d)$ is given by

(1.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}(d)<\frac{1}{2(d-1)}, & & \displaystyle\end{eqnarray}$$

an (almost) optimal Planck-scale result for $d=2$ , yet somewhat weaker than Berry’s conjecture for $d>2$ .

One can express the real toral Laplace eigenfunctions explicitly as a sum of exponentials

(1.5) $$\begin{eqnarray}\displaystyle f_{n}(x)=\mathop{\sum }_{\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}}c_{\unicode[STIX]{x1D706}}e(\langle x,\unicode[STIX]{x1D706}\rangle )\quad (c_{-\unicode[STIX]{x1D706}}=\overline{c_{\unicode[STIX]{x1D706}}}) & & \displaystyle\end{eqnarray}$$

for

(1.6) $$\begin{eqnarray}\displaystyle n\in S_{d}:=\{n=a_{1}^{2}+\cdots +a_{d}^{2}:\,a_{1},\ldots ,a_{d}\in \mathbb{Z}\} & & \displaystyle\end{eqnarray}$$

expressible as a sum of $d$ integer squares, and the corresponding frequencies $\unicode[STIX]{x1D706}$ are the standard lattice points

(1.7) $$\begin{eqnarray}\displaystyle {\mathcal{E}}_{n}={\mathcal{E}}_{d;n}=\{\unicode[STIX]{x1D706}\in \mathbb{Z}^{d}:\,\Vert \unicode[STIX]{x1D706}\Vert ^{2}=n\} & & \displaystyle\end{eqnarray}$$

lying on the $(d-1)$ -dimensional sphere (a circle for $d=2$ ) of radius $\sqrt{n}$ ; in this case the energy is $E=E_{n}=4\unicode[STIX]{x1D70B}^{2}n$ . We will assume without loss of generality that $f_{n}$ is $L^{2}$ -normalized, equivalent to

(1.8) $$\begin{eqnarray}\displaystyle \Vert f_{n}\Vert _{L^{2}(\mathbb{T}^{d})}^{2}=\mathop{\sum }_{\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}}|c_{\unicode[STIX]{x1D706}}|^{2}=1. & & \displaystyle\end{eqnarray}$$

For every $n\in S_{d}$ , denote

(1.9) $$\begin{eqnarray}\displaystyle N=N_{d;n}=\#{\mathcal{E}}_{n}. & & \displaystyle\end{eqnarray}$$

When $d=2$ , by Landau’s theorem, $\{n\leqslant x:\,n\in S_{2}\}\sim K(x/\sqrt{\log x})$ , where $K>0$ is the “Landau–Ramanujan constant”. On average $N=N_{2;n}$ is of order of magnitude $\sqrt{\log n}$ ; however, for a density-1 sequence in $S_{2}$ we have $N=(\log n)^{\log 2/2+o(1)}.$ In general, for $n\in S_{2}$ we have

$$\begin{eqnarray}N=n^{o(1)}.\end{eqnarray}$$

For $d=3$ , Siegel’s theorem asserts that for $n\not \equiv 0,4,7\,(8)$ ,

$$\begin{eqnarray}N=N_{3;n}=n^{1/2+o(1)};\end{eqnarray}$$

since $x\mapsto 2^{a}x$ is a bijection between the solutions to $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=n$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=4^{a}n$ , we can always assume that $n\not \equiv 0,4,7\,(8)$ with no loss of generality.

Granville and Wigman [Reference Granville and Wigman14, Theorem 1.2] refined the aforementioned estimate by Lester–Rudnick for $d=2$ . They proved that in this case, (1.3) is valid slightly above the Planck scale $r_{0}(E)=E^{-1/2+o(1)}$ for all eigenfunctions $f_{n}$ as in (1.5), corresponding to numbers $n$ so that the lattice points ${\mathcal{E}}_{n}$ are well separated (“Bourgain–Rudnick sequences”), a condition satisfied [Reference Bourgain and Rudnick7, Lemma 5] by “generic” integers $n\in S_{2}$ in a strong quantitative sense, subsequently refined in [Reference Granville and Wigman14, Theorem 1.4]; see §2.2.

1.3 Averaging mass with respect to ball centre

For both the two-dimensional and the higher-dimensional tori it is possible to construct exceptional examples of sequences of toral eigenfunctions where the equidistribution condition is not satisfied: for $d\geqslant 2$ thin sequences [Reference Lester and Rudnick23, Theorem 3.1] $\{\unicode[STIX]{x1D719}_{j_{k}\!}\}$ of eigenfunctions violating condition (1.2) at the Planck scale $r\cdot E_{\!j_{k}}^{1/2}\rightarrow \infty$ , around the origin $x=0$ , and even stronger, for $d\geqslant 3$  [Reference Lester and Rudnick23, Theorem 4.1 (construction by J. Bourgain)] eigenfunctions violating (1.2) with $r\gg E^{-\unicode[STIX]{x1D6FC}(d)}$ , where $\unicode[STIX]{x1D6FC}(d)>1/2(d-1)$ , again around the origin $x=0$ . In these cases, rather than keeping the ball centre $x=0$ at the origin, one may vary $x$ and study whether the “typical” discrepancy on the left-hand side of (1.2) is small, even if the existence of $x$ so that the left-hand side of (1.2) is not small is known, so that, in particular, (1.3) is not satisfied.

A natural way to vary $x$ is to think of $x$ as random, drawn uniformly in $\mathbb{T}^{d}$ . We define the random variable

(1.10) $$\begin{eqnarray}\displaystyle X_{\!f_{n},r}=X_{\!f_{n},r;x}:=\int _{B_{x}(r)}f_{n}(y)^{2}\,dy & & \displaystyle\end{eqnarray}$$

and are interested in the distribution of $X_{\!f_{n},r}$ where $x$ is drawn randomly uniformly in $\mathbb{T}^{d}$ . The relevant moments are: expectation

(1.11) $$\begin{eqnarray}\displaystyle \mathbb{E}[X_{\!f_{n},r}]=\int _{\mathbb{T}^{d}}X_{\!f_{n},r;x}\,dx, & & \displaystyle\end{eqnarray}$$

higher centred moments

(1.12) $$\begin{eqnarray}\displaystyle \mathbb{E}[(X_{\!f_{n},r}-\mathbb{E}[X_{\!f_{n},r}])^{k}]=\int _{\mathbb{T}^{d}}(X_{\!f_{n},r;x}-\mathbb{E}[X_{\!f_{n},r}])^{k}\,dx,\quad k\geqslant 2, & & \displaystyle\end{eqnarray}$$

and in particular the variance

(1.13) $$\begin{eqnarray}\displaystyle {\mathcal{V}}(X_{\!f_{n},r})=\mathbb{E}[(X_{\!f_{n},r}-\mathbb{E}[X_{\!f_{n},r}])^{2}]. & & \displaystyle\end{eqnarray}$$

This approach of averaging the $L^{2}$ -mass with respect to the ball centre (and keeping $f_{n}$ fixed) was pursued by Granville–Wigman [Reference Granville and Wigman14] in the two-dimensional case, again slightly above the Planck scale $r>E^{-1/2+o(1)}$ . In this regime, by proving an upper bound for ${\mathcal{V}}(X_{\!f_{n},r})$ beyond $(\mathbb{E}[X_{\!f_{n},r}])^{2}=O(r^{4})$ , valid for all $n\in S_{2}$ , under some flatness assumption on $f_{n}$ (cf. Definition 1.4 below), they established (1.2) for ‘‘typical” if not all $x\in \mathbb{T}^{2}$ . It would be desirable to find a regime where it is possible to analyse the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{\!f_{n},r})$ of $X_{\!f_{n},r}$ and, if possible, determine the limit distribution law for $X_{\!f_{n},r}$ ; our principal results below achieve both of these in the two-dimensional case and the former in the three-dimensional one (see Theorems 1.1 and 1.3). Such an approach of bounding the discrepancy variance while averaging over ball centres was recently used by Sarnak [Reference Sarnak25] for mass distribution of forms on symmetric spaces and Humphries [Reference Humphries20] for mass distribution of automorphic forms.

1.4 Statement of the main results for $d=2,3$ : asymptotics for the variance, central limit theorem

Our principal results below are applicable to “flat” functions for $d=2,3$ , understood in suitable, more and less restrictive, senses. For example, “Bourgain’s eigenfunction” [Reference Bourgain6]

(1.14) $$\begin{eqnarray}\displaystyle f_{n}(x)=\frac{1}{\sqrt{N}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}}\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D706}}e(\langle x,\unicode[STIX]{x1D706}\rangle ) & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D706}}=\pm 1$ for every $\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}$ , i.e., corresponding to $|c_{\unicode[STIX]{x1D706}}|=N^{-1/2},$ satisfies any of the flatness conditions in the most restrictive sense. Denote ${\mathcal{B}}_{n}$ to be the class of Bourgain’s eigenfunctions.

Our first principal result determines the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{\!f_{n},r})$ for the two-dimensional case and moreover asserts that the moments of the standardized random $L^{2}$ -mass of $f_{n}$ are asymptotically Gaussian; we subsequently deduce a central limit theorem (see Corollary 1.2). For the sake of elegance of presentation, it is formulated for Bourgain’s eigenfunctions (1.14); below we formulate a more general result which holds for a larger class of flat eigenfunctions (see Theorem 2.5 in §2) and later a result where the averaging over the ball centre $x$ is itself restricted to shrinking balls (Theorem 8.3 in §8).

The claimed uniform asymptotics (1.15) of the variance means explicitly that, as $n\rightarrow \infty$ along $S_{2}^{\prime }$ , one has

(1.19) $$\begin{eqnarray}\displaystyle \sup _{\substack{ r_{0}<r<(\log n)^{(1/2)\log (\unicode[STIX]{x1D70B}/2)-\unicode[STIX]{x1D716}} \\ f_{n}\in {\mathcal{B}}_{n}}}\biggl|\frac{{\mathcal{V}}(X_{\!f_{n},r})}{(16/3\unicode[STIX]{x1D70B})r^{4}T^{-1}}-1\biggr|\rightarrow 0 & & \displaystyle\end{eqnarray}$$

and the uniform convergence (1.18) of the moments means that for every $k\geqslant 3$ ,

$$\begin{eqnarray}\sup _{\substack{ r_{0}<r<r_{1} \\ f_{n}\in {\mathcal{B}}_{n}}}|\mathbb{E}[\hat{X}_{f_{n},r}^{k}]-\mathbb{E}[Z^{k}]|\rightarrow 0.\end{eqnarray}$$

Concerning the restricted range (1.16) in Theorem 1.1 (and (1.19)) for the possible radii, it is directly related to a well-known result on the angular distribution of lattice points in ${\mathcal{E}}_{n}$ for generic $n\in S_{2}$ . Namely, it was shown [Reference Erdős and Hall11] that ${\mathcal{E}}_{n}$ , projected by homothety to the unit circle, is equidistributed and, moreover, a quantitative measure for the discrepancy is asserted (see §2.1 below and, in particular, (2.2)), satisfied by generic $n\in S_{2}$ . Bourgain [Reference Bourgain6] observed that $f_{n}\in {\mathcal{B}}_{n}$ , when averaged over $x\in \mathbb{T}^{d}$ , exhibits Gaussianity in the following sense. Let $T>0$ be a fixed number and define the scaled function $\unicode[STIX]{x1D711}_{x}:[-1,1]^{2}\rightarrow \mathbb{R}$ around $x$ as

(1.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}_{x}(y):=f_{n}\bigg(x+\frac{T}{\sqrt{n}}\cdot y\bigg), & & \displaystyle\end{eqnarray}$$

i.e., the trace of $f_{n}$ on the side- $2(T/\sqrt{n})$ square centred at $x$ . It was found [Reference Bourgain6] that, upon thinking of $x\in \mathbb{T}^{2}$ as random, and $\unicode[STIX]{x1D711}_{x}(\cdot )$ as a random field indexed by $[-1,1]^{2}$ , it converges, in a suitable sense, to a particular Gaussian field (“monochromatic isotropic waves”) on $\mathbb{R}^{2}$ , restricted to $[-1,1]^{2}$ . This allows one to infer some results on the (deterministic) functions $f_{n}\in {\mathcal{B}}_{n}$ from the analogous results on the limit Gaussian random field. We may then re-interpret the quantitative version (2.2) of the angular equidistribution of lattice points as allowing the parameter $T$ in (1.20) to grow as a (positive) logarithmic power of $n$ , while still retaining the said asymptotic Gaussianity, also allowing for the comparison between the mass distribution of $f_{n}$ with respect to the position and mass distribution of monochromatic isotropic waves. Our intuition regarding the possibility of carrying on the explained “de-randomization” argument for establishing results of similar nature to the presented results was recently validated by Sartori [Reference Sartori26].

An application of the standard theory [Reference Feller12, §XVI.3, Lemma 2] allows us to infer a uniform central limit theorem for the random variables $\hat{X}_{\!f_{n},r}$ from the convergence (1.18) of their respective moments to the Gaussian ones.

Corollary 1.2. In the setting of Theorem 1.1, part (2), the distribution of the random variables $\{\hat{X}_{\!f_{n},r}\}$ converges uniformly to the standard Gaussian distribution: as $n\rightarrow \infty$ along $S_{2}^{\prime }$ ,

$$\begin{eqnarray}\operatorname{meas}\{x\in \mathbb{T}^{2}:\,\hat{X}_{\!f_{n},r;x}\leqslant t\}\rightarrow \frac{1}{\sqrt{2\unicode[STIX]{x1D70B}}}\int _{-\infty }^{t}\text{e}^{-z^{2}/2}\,dz\end{eqnarray}$$

uniformly for $t\in \mathbb{R}$ , $r_{0}<r<r_{1}$ and $f_{n}\in {\mathcal{B}}_{n}$ .

For the three-dimensional case, for Bourgain’s eigenfunctions, we only claim a precise asymptotic result on ${\mathcal{V}}(X_{\!f_{n},r})$ , the good news being that the claimed result is valid for all energies satisfying the natural congruence assumptions.

Theorem 1.3 (Asymptotics for the variance for $d=3$ , Bourgain’s eigenfunctions).

There exists a number $\unicode[STIX]{x1D702}>0$ such that if $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow \infty$ , then for all $n\not \equiv 0,4,7\,(8)$ we have

$$\begin{eqnarray}{\mathcal{V}}(X_{\!f_{n},r})\sim r^{6}T^{-2}\end{eqnarray}$$

uniformly for $r_{0}<r<n^{-1/2+\unicode[STIX]{x1D702}}$ and $f_{n}\in {\mathcal{B}}_{n}$ .

The meaning of the uniform statement in Theorem 1.3 is that

(1.21) $$\begin{eqnarray}\displaystyle \sup _{\substack{ r_{0}<r<n^{-1/2+\unicode[STIX]{x1D702}} \\ f_{n}\in {\mathcal{B}}_{n}}}\biggl|\frac{{\mathcal{V}}(X_{\!f_{n},r})}{r^{6}T^{-2}}-1\biggr|\rightarrow 0 & & \displaystyle\end{eqnarray}$$

as $n\rightarrow \infty$ along $n\not \equiv 0,4,7\,(8)$ ; cf. (1.19) in the two-dimensional case.

1.5 Statement of the main results for $d=2,3$ : more general upper and lower bounds

Let $f_{n}$ be as in (1.5) and consider the vector

(1.22) $$\begin{eqnarray}\displaystyle \text{}\underline{v}:=(|c_{\unicode[STIX]{x1D706}}|^{2})_{\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}}\in \mathbb{R}^{{\mathcal{E}}_{n}} & & \displaystyle\end{eqnarray}$$

of the squared absolute values of its coefficients; we denote its normalized $\ell _{\infty }$ -norm by

(1.23) $$\begin{eqnarray}\displaystyle [\text{}\underline{v}]_{\infty }:=N\cdot \max _{\unicode[STIX]{x1D706}\in {\mathcal{E}}_{n}}|c_{\unicode[STIX]{x1D706}}|^{2}. & & \displaystyle\end{eqnarray}$$

Definition 1.4 (Ultra-flat functions [Reference Granville and Wigman14, Definition 1.9]).

We say that an eigenfunction $f_{n}$ in (1.5) is $\unicode[STIX]{x1D716}$ -ultra-flat if its coefficients satisfy

(1.24) $$\begin{eqnarray}\displaystyle [\text{}\underline{v}]_{\infty }\leqslant N^{\unicode[STIX]{x1D716}}. & & \displaystyle\end{eqnarray}$$

Denote ${\mathcal{U}}_{n;\unicode[STIX]{x1D716}}$ to be the class of $\unicode[STIX]{x1D716}$ -ultra-flat functions.

The following couple of theorems establish more general upper and lower bounds on ${\mathcal{V}}(X_{\!f_{n},r})$ in the two- and three-dimensional cases, respectively.

Theorem 1.5 (Bounds for the variance for ultra-flat eigenfunctions, $d=2$ ).

There exist a density-1 sequence $S_{2}^{\prime }\subseteq S_{2}$ and an absolute constant $C>0$ such that for every $\unicode[STIX]{x1D716}>0$ , $\unicode[STIX]{x1D702}>0$ , $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow \infty$ arbitrarily slowly and $r=n^{-1/2}T>r_{0}$ , as $n\rightarrow \infty$ along $S_{2}^{\prime }$ , we have

(1.25) $$\begin{eqnarray}\displaystyle T^{-1}N^{-2\unicode[STIX]{x1D716}}\ll \frac{{\mathcal{V}}(X_{\!f_{n},r})}{r^{4}}\ll N^{\unicode[STIX]{x1D716}}\cdot (T^{-1}+(\log n)^{-(1/2)\log (\unicode[STIX]{x1D70B}/2)+\unicode[STIX]{x1D702}}) & & \displaystyle\end{eqnarray}$$

uniformly for $r_{0}<r<Cn^{-1/2}N^{1-\unicode[STIX]{x1D716}}$ and $f_{n}\in {\mathcal{U}}_{n;\unicode[STIX]{x1D716}}$ , where the constant involved in the “ $\ll$ ”-notation in (1.25) is absolute for the lower bound and depends only on $\unicode[STIX]{x1D702}$ for the upper bound. Moreover, the upper bound is valid for the extended range $r>r_{0}$ (with no upper bound on $r$ imposed) and the lower bound is valid for every $n\in S_{2}$ .

Theorem 1.6 (Bounds for the variance for ultra-flat eigenfunctions, $d=3$ ).

There exist a number $\unicode[STIX]{x1D702}>0$ and a constant $C>0$ such that for every $\unicode[STIX]{x1D716}>0$ , $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow \infty$ arbitrarily slowly, $r=n^{-1/2}T>r_{0}$ and $n\not \equiv 0,4,7\,(8)$ , we have

(1.26) $$\begin{eqnarray}\displaystyle T^{-2}N^{-2\unicode[STIX]{x1D716}}\ll \frac{{\mathcal{V}}(X_{\!f_{n},r})}{r^{6}}\ll N^{\unicode[STIX]{x1D716}}(T^{-2}+n^{-\unicode[STIX]{x1D702}}) & & \displaystyle\end{eqnarray}$$

uniformly for $r_{0}<r<Cn^{-1/2}N^{1-\unicode[STIX]{x1D716}}$ and $f_{n}\in {\mathcal{U}}_{n;\unicode[STIX]{x1D716}}$ , where the constants involved in the “ $\ll$ ”-notation are absolute. Moreover, the upper bound in (1.26) is valid for the extended range $r>r_{0}$ .

For Bourgain’s eigenfunctions, the proofs of Theorems 1.5 and 1.6 yield slightly stronger bounds compared to (1.25) and (1.26), namely

$$\begin{eqnarray}T^{-1}\ll \frac{{\mathcal{V}}(X_{\!f_{n},r})}{r^{4}}\ll T^{-1}+(\log n)^{-(1/2)\log (\unicode[STIX]{x1D70B}/2)+\unicode[STIX]{x1D716}}\end{eqnarray}$$

for $d=2$ and

$$\begin{eqnarray}T^{-2}\ll \frac{{\mathcal{V}}(X_{\!f_{n},r})}{r^{6}}\ll T^{-2}+n^{-\unicode[STIX]{x1D702}}\end{eqnarray}$$

for $d=3$ .

1.6 Outline of the paper

The rest of the paper is organized as follows. In §2 we formulate Theorem 2.5, which, on one hand, generalizes Theorem 1.1 for a larger class of flat eigenfunctions and, on the other hand, explicates a sufficient condition on $n\in S_{2}$ for its statements to hold; a few examples of application of Theorem 2.5, corresponding to different asymptotic behaviours of the variance (2.14), are also discussed. Section 4 is dedicated to giving a proof of the first part of Theorem 1.1 (respectively the first part of Theorem 2.5), whereas the second part of Theorem 1.1 (respectively the second part of Theorem 2.5) is proved in §5. Theorem 1.3, claiming the precise asymptotics for the $L^{2}$ -mass variance for Bourgain’s eigenfunctions in three dimensions, is proved in §6.

In §7 we prove the various upper and lower bounds asserted by Theorems 1.5 and 1.6. A refinement of Theorem 2.5, where rather than draw $x$ with respect to the uniform measure on the full torus, $x$ is drawn on balls slightly above the Planck scale, is presented in §8 and the additional subtleties of its proof as compared to the proof of Theorem 2.5 are highlighted. Finally, §9 contains the proofs of all auxiliary lemmas, postponed in course of the proofs of the various results.