Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus

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.


Nodal intersections and lattice points
Consider on the torus T 2 = R 2 /Z 2 a real-valued eigenfunction of the Laplacian F : T 2 → R, with eigenvalue λ 2 : (1.1) The nodal set of F is the zero locus {x ∈ T 2 : F(x) = 0}.
Let C ⊂ T 2 be a straight line segment on the torus, of length L: with |α| = 1 and 0 ≤ t ≤ L. We are interested in the number of nodal intersections It is well-known [8] that m ∈ S if and only if m = 2 ν · p α 1 1 · · · p α h h · q 2β 1 1 · · · q 2β l l , where each p i ≡ 1 mod 4 and each q j ≡ 3 mod 4; moreover, for m ∈ S, Given an eigenvalue λ 2 = 4π 2 m of (1.1), the collection {e 2πi μ,x } μ∈E is a basis for the eigenspace. All the eigenfunctions corresponding to the eigenvalue 4π 2 m are with c μ Fourier coefficients. The dimension of the eigenspace is N m = r 2 (m).
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 C of length L on the torus to be Moreover, they gave precise asymptotics for the variance of Z against smooth curves with nowhere zero curvature C (assuming w.l.o.g. to have unit speed parametrisation γ : [0, L] → C): This asymptotic behaviour is non-universal: B C (E) depends both on C and on the angular distribution of the lattice points. It also follows that the normalised number of nodal intersections Z √ m is a r.v. with constant mean and vanishing variance (as m → ∞ along a sequence s.t. N m → ∞): therefore, its distribution is asymptotically concentrated at the mean value.

Statement of main results
We study the nodal intersections Z for straight line segments, the other extreme of the nowhere zero curvature setting. Recall that the expectation of Z is given by (1.5). p or increasing products of any bounded number of primes (at least two of them), for example m k = (5 · 13) k .
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 = √ m. It was proven by Jarnik [9] that on every arc of length < ( √ m) 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]

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 C, we reduce the problem of studying the variance to bounding the second moment of the covariance function (2.2) below) and a couple of its derivatives. Next, using the hypothesis that 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 C is a line segment, so that the process f (t) = F(γ (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 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 α is the direction of our straight line. In Sect. 3, we bound (1.7) for α 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.

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 C to be a smooth toral curve (which may or may not be a segment). Let γ (t) : [0, L] → T 2 be its arc-length parametrisation. We restrict F along 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 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 → R are given by the Kac-Rice formulas (see [7], §10, and [1], We define the zero density function the latter defined for t 1 = t 2 . The Kac-Rice formulas for the first and second (factorial) moments of the number of crossings are Rudnick and Wigman proved that [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 ) ∈ [0, L]×[0, L] such that t 1 = 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).
This result is applicable to the case where 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 C. From this point on, assume C ⊂ T 2 to be a segment; we write with |α| = 1 and 0 ≤ t ≤ 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 C is a segment, then the process f is stationary (and without loss of generality we may assume that 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 ∈ R 2 , we define the set with E as in (1.3).

Proposition 2.3
Assuming C to be a segment, The proof of Proposition 2.3 is given in Sect. 6.

The case of rational lines
The goal of this section is to prove Theorem 1.1. Recall that E = {μ ∈ Z 2 : |μ| 2 = m} is the set of lattice points lying on the circle of radius √ m, and N m = |E| is their number. By Proposition 2.3, it is sufficient to bound the summation

and A α be as in Definition 2.2. Then
Proof Up to multiplication by a scalar, α has integer coordinates: for some p, q ∈ Z and q = 0. Note that A α = A (q, p) because the vectors α and (q, p) are collinear. It follows that (3.2) Next, let μ be fixed, and consider k = μ − μ , (q, p) . As both μ − μ and (q, p) have integer coordinates, k ∈ Z; moreover, as (μ, μ ) ∈ A (q, p) , k = 0. Then We now show that there can be at most two terms in the inner-most summation: the lattice point μ of the circle x 2 + y 2 = m has to satisfy, for fixed μ and k, Thus μ is lying on the straight line qx + py = h, and a circle and a line can intersect in at most two points. Therefore,

Lattice points on short arcs
The number of lattice points N m on the circle of radius √ m has the upper bound N m m ∀ > 0, (4.1) and the analogous statement with powers of logarithms of m in place of m 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 < ( √ m) 1 3 of the circle there are at most 2 lattice points [9]. Moreover, Cilleruelo and Córdoba [4] proved that, for all integers l ≥ 1, on any arc of length ≤ √ 2( √ m)    It is known [11] that, as X → ∞, S(X ) ∼ C X √ log X , where C > 0 is the Landau-Ramanujan constant.

Lemma 4.3 (Bourgain and Rudnick [2], Lemma 5) Fix > 0. Then for all but O(X 1− 3 ) integers m ≤ X , one has
Therefore, the assumptions of Theorem 1.5 hold for a density one sequence of energy levels.

The case of irrational lines
The goal of this section is to prove Theorems 1.2, 1.4 and 1.5.

Preparatory results
Denote √ mS 1 the radius √ m circle.
we have Let • In case E lies on s between B and E , we have where we have denoted O the origin, centre of √ mS 1 .
• In case E lies on s between B and E , then B lies on the arc DE and we have • In case E = B, we write For two functions f (m), g(m), we write f ∼ g if, as m → ∞, the ratio of the two sides converges to 1.

Proposition 5.2
Let A α be as in Definition 2.2, and recall that |α| = 1. Assume that every arc on √ mS 1 of length J contains at most l lattice points. Then Proof Let a ≤ 2 √ m and c be positive parameters, such that c → 0 as m → ∞. We separate the sum over the following three ranges: We may now rewrite First range: recall the notation √ mS 1 for the radius √ m circle. For a fixed lattice point μ, all μ satisfying |μ − μ | ≤ a must lie on a disc centred at μ with radius a; the intersection of this disc with √ mS 1 is an arc on √ mS 1 of length ∼ a around μ. To bound (from above) the number of μ on this arc, we partition it into small arcs of length J : there are 1 + a J small arcs, and by the assumptions of Proposition 5.2 each contains at most l lattice points. Therefore, Third range: Here we have |μ − μ | ≥ a and | μ − μ , α | ≥ c|μ − μ |, therefore The optimal choices for the parameters are Proof By Conjecture 1.3, for some > 0, we may take J = ( √ m) where the latter inequality follows from (4.1).
Proof By the assumptions of Corollary 5.5, we have that on the circle √ mS 1 on any arc of length < ( √ m) 1− there is at most one lattice point. Therefore, we may take J = ( √ m) 1− and l = 1 in Proposition 5.2, yielding where the latter inequality follows from (4.1).

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 C, and the notation Also recall the definition (2.5) of R 2 (m). We begin by squaring the covariance function (2.6):

Lemma 6.1 Let C be a segment. Then
yielding (6.1). Next, and it follows that By Cauchy-Schwartz, L 0 e 2πit μ−μ ,α dt 2 and (6.2) follows. For the second mixed derivative: Proof We split the summation over three ranges: diagonal pairs, off-diagonal pairs satisfying μ − μ ⊥ α, and the set A α of Definition 2.2: (6.4) The sum for μ = μ 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 μ − μ ⊥ α, 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: The result follows on replacing (6.5), (6.6) and (6.9) into (6.4).
Proof of Proposition 2.3 By Proposition 2.1, Lemma 6.1 and Lemma 6.2: