Abstract
We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. We found an upper bound for the nodal intersections variance, depending on whether the slope of the straight line is rational or irrational. Our findings exhibit a close relation between this problem and the theory of lattice points on circles.
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1 Introduction
1.1 Nodal intersections and lattice points
Consider on the torus \(\mathbb {T}^2=\mathbb {R}^2/\mathbb {Z}^2\) a real-valued eigenfunction of the Laplacian \(F:\mathbb {T}^2\rightarrow \mathbb {R}\), with eigenvalue \(\lambda ^2\):
The nodal set of F is the zero locus
Let \(\mathcal {C}\subset \mathbb {T}^2\) be a straight line segment on the torus, of length L:
with \(|\alpha |=1\) and \(0\le t\le L\). We are interested in the number of nodal intersections
the number of zeros of F on \(\mathcal {C}\), as \(\lambda \rightarrow \infty \).
This problem is closely related to the theory of lattice points on circles, as we shall now see. The sequence of Laplace eigenvalues, or energy levels, on \(\mathbb {T}^2\) is given by
where \(S:=\{m: \ m=a^2+b^2, a,b\in \mathbb {Z}\}\). For \(m\in S\), let
be the set of all lattice points on the circle of radius \(\sqrt{m}\). The number \(|\mathcal {E}|\) of lattice points equals \(r_2(m)\), the number of ways to write m as sum of two integer squares. We shall denote
It is well-known [8] that \(m\in S\) if and only if \(m=2^{\nu }\cdot p_1^{\alpha _1}\cdots p_h^{\alpha _h}\cdot q_1^{2\beta _1}\cdots q_l^{2\beta _l}\), where each \(p_i\equiv 1 \mod 4\) and each \(q_j\equiv 3 \mod 4\); moreover, for \(m\in S\),
Given an eigenvalue \(\lambda ^2=4\pi ^2m\) of (1.1), the collection \(\{e^{2\pi i\langle \mu ,x\rangle }\}_{\mu \in \mathcal {E}}\) is a basis for the eigenspace. All the eigenfunctions corresponding to the eigenvalue \(4\pi ^2m\) are
with \(c_{\mu }\) Fourier coefficients. The dimension of the eigenspace is \(N_m=r_2(m)\).
1.2 The model and prior results
We consider the random Gaussian toral eigenfunctions, called “arithmetic random waves” [10]
where \(a_{\mu }\) are complex standard Gaussian random variables (i.e. \(\mathbb {E}(a_{\mu })=0\) and \(\mathbb {E}(|a_{\mu }|^2)=1\)), independent save for the relations \(a_{-\mu }=\overline{a_{\mu }}\) (so that F(x) is real-valued).
One is interested in the distribution of the nodal intersections (1.2). Rudnick and Wigman [12] computed the expected number of nodal intersections against smooth curves \(\mathcal {C}\) of length L on the torus to be
Moreover, they gave precise asymptotics for the variance of \(\mathcal {Z}\) against smooth curves with nowhere zero curvature \(\mathcal {C}\) (assuming w.l.o.g. to have unit speed parametrisation \(\gamma : [0,L]\rightarrow \mathcal {C}\)):
where
This asymptotic behaviour is non-universal: \(B_{\mathcal {C}}(\mathcal {E})\) depends both on \(\mathcal {C}\) and on the angular distribution of the lattice points. It also follows that the normalised number of nodal intersections \(\frac{\mathcal {Z}}{\sqrt{m}}\) is a r.v. with constant mean and vanishing variance (as \(m\rightarrow \infty \) along a sequence s.t. \(N_m\rightarrow \infty \)): therefore, its distribution is asymptotically concentrated at the mean value.
1.3 Statement of main results
We study the nodal intersections \(\mathcal {Z}\) for straight line segments, the other extreme of the nowhere zero curvature setting. Recall that the expectation of \(\mathcal {Z}\) is given by (1.5).
Theorem 1.1
Let \(\mathcal {C}\subset \mathbb {T}^2\) be a length L segment with rational slope, i.e. \(\gamma (t)=t\alpha \), \(\alpha =(\alpha _1,\alpha _2)\) with \(\frac{\alpha _2}{\alpha _1}\in \mathbb {Q}\), \(|\alpha |=1\), and \(\{m\}\subseteq S\) a sequence such that \(N_m\rightarrow \infty \). Then
the implied constant depending on \(\alpha \) only.
This upper bound for the variance is the same order of magnitude as the leading term in (1.6) for the case of nowhere zero curvature.
Without the assumption of rational slope we may prove the following result unconditionally.
Theorem 1.2
Let \(\mathcal {C}\) be a segment on the torus, and \(\{m\}\subseteq S\) a sequence such that \(N_m\rightarrow \infty \). Then
The variance of \(\frac{\mathcal {Z}}{\sqrt{m}}\) vanishes for all sequences \(\{m\}\subseteq S\) satisfying
Examples of such sequences include increasing products of distinct primes
or increasing products of any bounded number of primes (at least two of them), for example
We may improve the bound of Theorem 1.2 conditionally on a conjecture about lattice points on short arcs. Consider a circle of radius \(R=\sqrt{m}\). It was proven by Jarnik [9] that on every arc of length \(<(\sqrt{m})^\frac{1}{3}\) there are at most 2 lattice points. Theorem 1.4 below is conditional on a weaker version of a conjecture by Cilleruelo and Granville (Conjecture 4.2 in Sect. 4; see also [5, 6]).
Conjecture 1.3
There exists \(\epsilon >0\) such that on a circle of radius \(R=\sqrt{m}\), on any arc of length \((\sqrt{m})^{\frac{1}{2}+\epsilon }\) there are O(1) lattice points.
Theorem 1.4
Assume Conjecture 1.3. Let \(\mathcal {C}\) be a segment on the torus, and \(\{m\}\subseteq S\) a sequence such that \(N_m\rightarrow \infty \). Then
Furthermore, we may prove the bound of Theorem 1.4 unconditionally for a density one sequence of energy levels (cf. Lemma 4.3).
Theorem 1.5
Let \(\mathcal {C}\) be a segment on the torus, and \(\{m\}\subseteq S\) a sequence such that \(N_m\rightarrow \infty \) and
for some \(0<\epsilon <\frac{1}{2}\) and sufficiently big m. Then
1.4 Outline of the paper
The rest of this work focuses on proving the stated theorems. In Sect. 2, thanks to the work of Rudnick and Wigman [12] for generic curves \(\mathcal {C}\), we reduce the problem of studying the variance to bounding the second moment of the covariance function \(r(t_1,t_2)=\mathbb {E}[F(\gamma (t_1))F(\gamma (t_2))]\) (see (2.2) below) and a couple of its derivatives. Next, using the hypothesis that \(\mathcal {C}\) is a segment, we further reduce our problem to bounding sums over the lattice points. This relies on estimates for the second moment (established in Sect. 6).
There are marked differences compared to the case of generic curves: first, the covariance function has the special form (2.6) if \(\mathcal {C}\) is a line segment, so that the process \(f(t)=F(\gamma (t))\) (see (2.1) below) is stationary. This leads to a different method from [12] of controlling the second moment, and specifically the off-diagonal terms of (6.4). Indeed, in [12], Lemma 5.2, the off-diagonal terms are handled via Van der Corput’s lemma, applicable for curves \(\mathcal {C}\) of nowhere vanishing curvature, whereas the special form (2.6) of the covariance function allows us to establish the estimate (6.7) directly; the latter term happens to be of different nature than the corresponding expression in the non-vanishing curvature case (cf. [12], Equation (5.18)). This leads to bounding a certain summation over the lattice points, different from [12]: Rudnick and Wigman proved that (see [12], Proposition 5.3)
whereas in this work, we need to bound
where \(\alpha \) is the direction of our straight line. In Sect. 3, we bound (1.7) for \(\alpha \) rational, and complete the proof of Theorem 1.1; in Sect. 5, we treat the case of irrational slope, and complete the proofs of Theorems 1.2, 1.4 and 1.5, following necessary background on the number of lattice points belonging to a short arc of a circle, covered in Sect. 4.
2 An approximate Kac–Rice formula
The random Gaussian toral eigenfunction (1.4) is a stationary Gaussian random field. Indeed, the covariance function is
depending on \(x-y\) only. The covariance function of a random field is non-negative definite (see [7], §5.1); a (centred) Gaussian random field is completely determined by its covariance function (see Kolmogorov’s Theorem [7], §3.3).
For now we assume \(\mathcal {C}\) to be a smooth toral curve (which may or may not be a segment). Let \(\gamma (t): [0,L]\rightarrow \mathbb {T}^2\) be its arc-length parametrisation. We restrict F along \(\mathcal {C}\), which yields the (centred Gaussian) random process f on the interval [0, L]:
Its covariance function is
The quantity we are studying, i.e. the number of nodal intersections \(\mathcal {Z}\), equals the number of zero crossings of the process f (on [0, L]). The moments of a random variable that counts the number of crossings of a level by a process \(f:I\rightarrow \mathbb {R}\) are given by the Kac–Rice formulas (see [7], §10, and [1], Theorem 3.2). For each t, let \(\phi _{f(t)}\) be the probability density function of the (standard Gaussian) random variable f(t), and \(\phi _{f(t_1),f(t_2)}\) the joint density of the random vector \((f(t_1),f(t_2))\). We define the zero density function \(K_1: [0,L]\rightarrow \mathbb {R}\) and 2-point correlation function \(K_2: [0,L]\times [0,L]\rightarrow \mathbb {R}\) of a process f as the Gaussian expectations
the latter defined for \(t_1\ne t_2\). The Kac–Rice formulas for the first and second (factorial) moments of the number of crossings are
Rudnick and Wigman proved that \(K_1(t)\equiv \sqrt{2}\sqrt{m}\) (see [12], Lemma 2.1), and via (2.3) they computed the expected intersection number (recall (1.5)).
The Kac–Rice formula for the second moment (2.4) holds provided the following non-degeneracy condition is met by f: the centred Gaussian distribution of the vector \((f(t_1),f(t_2))\) must be nondegenerate for all \((t_1,t_2)\in [0,L]\times [0,L]\) such that \(t_1\ne t_2\) (see [1], §3). This may fail for f as in (2.1); however, Rudnick and Wigman developed an approximate Kac–Rice formula. Denote
the derivatives of the covariance function (2.2).
Proposition 2.1
(Approximate Kac–Rice bound [13], Proposition 2.2) We have
where
This result is applicable to the case where \(\mathcal {C}\) is a segment, as it holds for all smooth curves. Note that the approximate Kac–Rice formula [12], Proposition 1.3 gives both the leading term and the error term for the variance; the upper bound of Proposition 2.1 is sufficient for our purposes. Our problem is thus reduced to bounding the second moment of the covariance function and a couple of its derivatives along \(\mathcal {C}\).
From this point on, assume \(\mathcal {C}\subset \mathbb {T}^2\) to be a segment; we write
with \(|\alpha |=1\) and \(0\le t\le L\). In this case, (2.1) becomes
and the covariance function of the process is
depending on the difference \(t_1-t_2\) only. Therefore, if \(\mathcal {C}\) is a segment, then the process f is stationary (and without loss of generality we may assume that \(\mathcal {C}\) contains the origin).
We now further reduce our problem to bounding a sum over the lattice points.
Definition 2.2
Given a nonzero vector \(v\in \mathbb {R}^2\), we define the set
with \(\mathcal {E}\) as in (1.3).
Proposition 2.3
Assuming \(\mathcal {C}\) to be a segment,
3 The case of rational lines
The goal of this section is to prove Theorem 1.1. Recall that
is the set of lattice points lying on the circle of radius \(\sqrt{m}\), and \(N_m=|\mathcal {E}|\) is their number. By Proposition 2.3, it is sufficient to bound the summation
Proposition 3.1
Let \(\alpha =(\alpha _1,\alpha _2)\) with \(\frac{\alpha _2}{\alpha _1}\in \mathbb {Q}\), and \(A_\alpha \) be as in Definition 2.2. Then
Proof
Up to multiplication by a scalar, \(\alpha \) has integer coordinates:
for some \(p,q\in \mathbb {Z}\) and \(q\ne 0\). Note that \( A_\alpha = A_{(q,p)} \) because the vectors \(\alpha \) and (q, p) are collinear. It follows that
Next, let \(\mu \) be fixed, and consider \(k=\langle \mu -\mu ',(q,p)\rangle \). As both \(\mu -\mu '\) and (q, p) have integer coordinates, \(k\in \mathbb {Z}\); moreover, as \((\mu ,\mu ')\in A_{(q,p)}\), \(k\ne 0\). Then
We now show that there can be at most two terms in the inner-most summation: the lattice point \(\mu '\) of the circle \(x^2+y^2=m\) has to satisfy, for fixed \(\mu \) and k,
Thus \(\mu '\) is lying on the straight line \(qx+py=h\), and a circle and a line can intersect in at most two points. Therefore,
Combining (3.2), (3.3) and (3.4) we get the statement (3.1) of Proposition 3.1. \(\square \)
Proof of Theorem 1.1
Applying Proposition 2.3, we have
with \(A_\alpha \) as in Definition 2.2. By Proposition 3.1,
and the statement of Theorem 1.1 follows. \(\square \)
4 Lattice points on short arcs
The number of lattice points \(N_m\) on the circle of radius \(\sqrt{m}\) has the upper bound
and the analogous statement with powers of logarithms of m in place of \(m^\epsilon \) is false [8]. We are interested in upper bounds for the number of lattice points on short arcs of the circle (the term ‘short’ indicates that the length of the arc is small compared to the radius): we now review the known bounds. As mentioned in the introduction, on any arc of length \(<(\sqrt{m})^\frac{1}{3}\) of the circle there are at most 2 lattice points [9].
Moreover, Cilleruelo and Córdoba [4] proved that, for all integers \(l\ge 1\), on any arc of length \(\le \sqrt{2}(\sqrt{m})^ {\frac{1}{2}-\frac{1}{(4\lfloor \frac{l}{2} \rfloor +2)}}\) there are at most l lattice points.
Proposition 4.1
(Bourgain and Rudnick [3], Lemma 2.1) On any arc of length at most \((\sqrt{m})^{\frac{1}{2}}\) of a circle of radius \(\sqrt{m}\), there are \(O(\log {m})\) lattice points.
Conjecture 4.2
(Cilleruelo and Granville [5, 6]) Consider a circle of radius \(R=\sqrt{m}\). For all \(\delta >0\), there exists a constant \(C_{\delta }\) such that on any arc of length \((\sqrt{m})^{1-\delta }\) there are at most \(C_{\delta }\) lattice points.
Conjecture 4.2 implies Conjecture 1.3. Furthermore, Bourgain and Rudnick [2] showed that Conjecture 4.2 is true for ‘most’ \(m\in S\). Recalling that \(S=\{m: \ m=a^2+b^2, a,b\in \mathbb {Z}\}\), define
It is known [11] that, as \(X\rightarrow \infty \), \(S(X)\sim C\frac{X}{\sqrt{\log X}}\), where \(C>0\) is the Landau-Ramanujan constant.
Lemma 4.3
(Bourgain and Rudnick [2], Lemma 5) Fix \(\epsilon >0\). Then for all but \(O(X^{1-\frac{\epsilon }{3}})\) integers \(m\le X\), one has
Therefore, the assumptions of Theorem 1.5 hold for a density one sequence of energy levels.
5 The case of irrational lines
The goal of this section is to prove Theorems 1.2, 1.4 and 1.5.
5.1 Preparatory results
Denote \(\sqrt{m}\mathcal {S}^1\) the radius \(\sqrt{m}\) circle.
Lemma 5.1
Let \(c=c(m)>0\), with \(c\rightarrow 0\) as \(m\rightarrow \infty \). Fix a point \(B\in \sqrt{m}\mathcal {S}^1\), and let \(\beta \) be a unit vector. Then there exists an arc \(\overset{\frown }{DE}\) of \(\sqrt{m}\mathcal {S}^1\) of length \((4c+O(c^3))\sqrt{m}\) such that all points \(B'\in \sqrt{m}\mathcal {S}^1\) satisfying \(B'\ne B\) and \(|\langle B-B',\beta \rangle |\le c|B-B'|\) lie on \(\overset{\frown }{DE}\).
Proof
The condition \(|\langle B-B',\beta \rangle |\le c|B-B'|\) means \(B-B'\) and \(\beta \) are close to being orthogonal, in the sense that \(|\cos (\varphi _{B-B',\beta })|\le c\), where \(0\le \varphi _{v,w}\le \pi \) denotes the angle between two non-zero vectors \(v,w\in \mathbb {R}^2\). Let \(s',s''\) be the two straight lines through B satisfying
Let D be the further intersection between the circle \(\sqrt{m}\mathcal {S}^1\) and \(s'\), meaning \(\sqrt{m}\mathcal {S}^1\cap s'=\{B,D\}\). Likewise, let E be the further intersection between \(\sqrt{m}\mathcal {S}^1\) and \(s''\), meaning \(\sqrt{m}\mathcal {S}^1\cap s''=\{B,E\}\). Note that possibly one of the lines \(s',s''\), say \(s''\), is tangent to the circle \(\sqrt{m}\mathcal {S}^1\), in which case \(E=B\). We have \(B'\in \overset{\frown }{DE}\) and we want to show \(\overset{\frown }{DE}=(4c+O(c^3))\sqrt{m}\).
By the expansion
we have
Let \(D',D''\) be points on \(s'\) on opposite sides of B, and \(E',E''\) be points on \(s''\) on opposite sides of B, so that: \(\overline{BD'}=\overline{BD''}=\overline{BE'}=\overline{BE''}=3\sqrt{m}\), D lies on \(s'\) between B and \(D'\), and \(\widehat{D'BE'}=\varphi _{s',s''}=2c+O(c^3)\). There are three cases:
-
In case E lies on \(s''\) between B and \(E'\), we have
$$\begin{aligned} \overset{{\displaystyle \frown }}{DE}=\widehat{DOE}\cdot \sqrt{m}=2\widehat{D'BE'}\cdot \sqrt{m}=(4c+O(c^3))\sqrt{m} \end{aligned}$$where we have denoted O the origin, centre of \(\sqrt{m}\mathcal {S}^1\).
-
In case E lies on \(s''\) between B and \(E''\), then B lies on the arc \(\overset{\frown }{DE}\) and we have
$$\begin{aligned} \overset{{\displaystyle \frown }}{DE}= & {} (\widehat{DOB}+\widehat{EOB})\sqrt{m}=(2\widehat{DEB}+2\widehat{EDB})\sqrt{m}=2\widehat{D'BE'}\cdot \sqrt{m} \\= & {} (4c+O(c^3))\sqrt{m}. \end{aligned}$$ -
In case \(E=B\), we write
$$\begin{aligned} \overset{{\displaystyle \frown }}{DE} = \overset{{\displaystyle \frown }}{DB} = \widehat{DOB}\cdot \sqrt{m} = 2\widehat{D'BE'}\cdot \sqrt{m} =(4c+O(c^3))\sqrt{m}. \end{aligned}$$
\(\square \)
For two functions f(m), g(m), we write \(f\sim g\) if, as \(m\rightarrow \infty \), the ratio of the two sides converges to 1.
Proposition 5.2
Let \(A_\alpha \) be as in Definition 2.2, and recall that \(|\alpha |=1\). Assume that every arc on \(\sqrt{m}\mathcal {S}^1\) of length J contains at most l lattice points. Then
Proof
Let \(a\le 2\sqrt{m}\) and c be positive parameters, such that \(c\rightarrow 0\) as \(m\rightarrow \infty \). We separate the sum over the following three ranges:
-
first range: \(|\mu -\mu '|\le a\)
-
second range: \(|\langle \mu -\mu ',\alpha \rangle |\le c|\mu -\mu '|\)
-
third range: \(|\mu -\mu '|\ge a, \ |\langle \mu -\mu ',\alpha \rangle |\ge c|\mu -\mu '|\).
We may now rewrite
First range: recall the notation \(\sqrt{m}\mathcal {S}^1\) for the radius \(\sqrt{m}\) circle. For a fixed lattice point \(\mu \), all \(\mu '\) satisfying \(|\mu -\mu '|\le a\) must lie on a disc centred at \(\mu \) with radius a; the intersection of this disc with \(\sqrt{m}\mathcal {S}^1\) is an arc on \(\sqrt{m}\mathcal {S}^1\) of length \(\sim a\) around \(\mu \). To bound (from above) the number of \(\mu '\) on this arc, we partition it into small arcs of length J: there are \(\ll 1+\frac{a}{J}\) small arcs, and by the assumptions of Proposition 5.2 each contains at most l lattice points. Therefore,
Second range: fix a lattice point \(\mu \) and apply Lemma 5.1 with \(\beta =\alpha \). Then all \(\mu '\) satisfying \(|\langle \mu -\mu ',\alpha \rangle |\le c|\mu -\mu '|\) must lie on an arc of length \((4c+O(c^3))\sqrt{m}\) on the circle \(\sqrt{m}\mathcal {S}^1\). Partition this arc into small arcs of length J: there are \(\ll 1+\frac{4c\sqrt{m}}{J}\) small arcs, and each contains at most l lattice points. It follows that
Third range: Here we have \(|\mu -\mu '|\ge a\) and \(|\langle \mu -\mu ',\alpha \rangle |\ge c|\mu -\mu '|\), therefore
Substituting (5.2), (5.3) and (5.4) into (5.1), we obtain
The optimal choices for the parameters are
and it follows that
\(\square \)
5.2 Proofs of Theorems 1.2, 1.4 and 1.5
Corollary 5.3
We have unconditionally
Proof
By Proposition 4.1, we may take \(J=(\sqrt{m})^{\frac{1}{2}}\) and \(l=O(\log (m))\) unconditionally in Proposition 5.2. \(\square \)
Proof of Theorem 1.2
Apply Proposition 2.3, yielding (3.5); by Corollary 5.3, we have
where we have assumed \(\log m=o(N_m)\). \(\square \)
Corollary 5.4
Assume Conjecture 1.3. Then
Proof
By Conjecture 1.3, for some \(\epsilon >0\), we may take \(J=(\sqrt{m})^{\frac{1}{2}+\epsilon }\) and \(l=O(1)\) in Proposition 5.2:
where the latter inequality follows from (4.1). \(\square \)
Proof of Theorem 1.4
Apply Proposition 2.3, yielding (3.5); by Corollary 5.4,
\(\square \)
Corollary 5.5
Let \(\{m\}\subseteq S\) be a sequence satisfying
for some \(0<\epsilon <\frac{1}{2}\) and sufficiently big m. Then
Proof
By the assumptions of Corollary 5.5, we have that on the circle \(\sqrt{m}\mathcal {S}^1\) on any arc of length \(<(\sqrt{m})^{1-\epsilon }\) there is at most one lattice point. Therefore, we may take \(J=(\sqrt{m})^{1-\epsilon }\) and \(l=1\) in Proposition 5.2, yielding
where the latter inequality follows from (4.1). \(\square \)
Proof of Theorem 1.5
Apply Proposition 2.3, yielding (3.5); by Corollary 5.5, we have
\(\square \)
6 The second moment of r and of its derivatives
In this section we prove Proposition 2.3, for which we need two auxiliary lemmas. Recall that \(r=r(t_1,t_2)\) is the covariance function restricted to \(\mathcal {C}\), and the notation
Also recall the definition (2.5) of \(\mathcal {R}_2(m)\).
Lemma 6.1
Let \(\mathcal {C}\) be a segment. Then
Proof
We will show
for \(i=1,2\), and
We begin by squaring the covariance function (2.6):
so that
yielding (6.1). Next,
and it follows that
By Cauchy–Schwartz,
and (6.2) follows. For the second mixed derivative:
thus
Again by Cauchy–Schwartz,
yielding (6.3). \(\square \)
Lemma 6.2
We have the following bound:
Proof
We split the summation over three ranges: diagonal pairs, off-diagonal pairs satisfying \(\mu -\mu '\perp \alpha \), and the set \(A_\alpha \) of Definition 2.2:
The sum for \(\mu =\mu '\) contains \(N_m\) summands (cf. [12], Sect. 5):
By Zygmund’s trick [14], there can be at most \(N_m\) pairs of lattice points satisfying \(\mu -\mu '\perp \alpha \), since on a circle there are at most two chords with given length and direction. Thus, the sum for this range contains at most \(N_m\) terms:
Given a summand
in the range \((\mu ,\mu ')\in A_\alpha \), we integrate and apply the triangle inequality:
Also by the triangle inequality,
The result follows on replacing (6.5), (6.6) and (6.9) into (6.4). \(\square \)
Proof of Proposition 2.3
By Proposition 2.1, Lemma 6.1 and Lemma 6.2:
\(\square \)
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Acknowledgments
This work was carried out as part of the author’s PhD thesis at King’s College London, under the supervision of Dr. Igor Wigman. The author’s PhD is funded by a Graduate Teaching Assistantship, Department of Mathematics. The author wishes to thank Dr. Igor Wigman for his invaluable guidance, remarks and corrections. The author wishes to thank Prof. Zeév Rudnick for suggesting this very interesting problem, and for helpful communications.
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Maffucci, R.W. Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus. Monatsh Math 183, 311–328 (2017). https://doi.org/10.1007/s00605-016-1001-2
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DOI: https://doi.org/10.1007/s00605-016-1001-2