ON THE GEOMETRY OF THE NODAL LINES OF EIGENFUNCTIONS

arXiv:1012.3843v2 [math.AP] 28 Jan 2011

ON THE GEOMETRY OF THE NODAL LINES OF EIGENFUNCTIONS OF THE TWO-DIMENSIONAL TORUS ´ RUDNICK JEAN BOURGAIN AND ZEEV Abstract. The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.

1. Introduction In this paper, we study the geometry of nodal lines of eigenfunctions of the Laplacian on the standard flat torus T2 = R2 /Z2 . The eigenvalues of the Laplacian on T2 are of the form 4π 2 E, where E = n21 + n22 is an integer which is a sum of two squares (in the sequel we will abuse notation and refer to E as the eigenvalue), the corresponding eigenspace being trigonometric polynomials of the form X (1.1) ϕ(x) = aξ e(x · ξ) ξ∈Z2 ,|ξ|2 =E

where we abbreviate e(t) := exp(2πit). In order for ϕ to be real-valued, the Fourier coefficients must satisfy aξ = a−ξ . Given the eigenfunction ϕ, we may consider its nodal set (1.2)

Nϕ = {ϕ = 0} .

According to Courant’s theorem, the complement of Nϕ has at most O(E) connected components, the “nodal domains”. Their boundary are the “nodal lines”. For any two-dimensional surface, it is known [Ch] that the nodal lines are a union of C 2 -immersed circles, with at most finitely many singular points and the nodal lines through a singular point form an equiangular system, see Figure 1. Thus, with the exception of the singular set, the nodal set of an eigenfunction is rectifiable and we can speak about its length. In the realanalytic case, such as in our case of the flat torus, Donnelly-Fefferman [D-F] showed that the length of the nodal set of an eigenfunction with eigenvalue Date: January 31, 2011. 1

´ RUDNICK JEAN BOURGAIN AND ZEEV

2

E = λ2 is commensurable to λ: length(Nϕ ) ≈

(1.3)

√

2Π

2Π

Π

Π

Π

E=λ.

2Π

Π

2Π

Figure 1. Nodal lines for the eigenfunction cos(4x − 7y) + sin(8x − y) + sin(4x + 7y) (left) and sin(4x + 7y) + sin(4x − 7y) + sin(8x + y) + sin(8x − y) = 2 sin 4x cos y(−1 + 2 cos 4x + 2 cos 2y − 2 cos 4y + 2 cos 6y) (right). Our goal in this paper is to better understand the local geometry of nodal lines. In this respect, M. Berry argued [Be] that for random plane waves, the nodal lines typically have curvature of order E. If one tries to make a statement for nodal lines of individual eigenfunctions, say in the case of T2 , it is clear that ‘pointwise curvature’ is not the appropriate concept. Indeed, nodal lines (for arbitrary large E) may have zero curvature or, as is also easily seen, develop arbitrary large pointwise curvature (for a fixed E). 1.1. The width of nodal lines. In order to formulate an alternative to curvature, we first introduce some terminology. Definition 1. An arc C ⊂ T2 is called ‘regular’ if C admits an arc-length parametrization γ : [0, ℓ] → C, 0 < ℓ < 1, which is C 2 and such that for some κ > 0, the curvature |¨ γ | satisfies a pointwise pinching condition (1.4)

..

κ < |γ| < 2κ

and the total curvature is bounded: (1.5)

2κℓ < 1 .

For a convex curve C the width w(C) is defined as the minimal distance between a pair of parallel supporting lines of the curve (Figure 2). In the case of regular arcs, we shall see that (1.6)

w(C) ≈ ℓ2 κ

3

wHCL

ON THE GEOMETRY OF THE NODAL LINES

l Figure 2. The width of a regular arc An examination of numerical plots of some nodal sets (see Figure 1) leads one to realize that Nϕ does not contain “large” curved arcs, specifically that as E → ∞, for regular arcs C ⊂ Nϕ , either their length ℓ → 0 shrinks or the curvature pinching κ → 0. We make the following conjecture: Conjecture 1. For all ε > 0, there is cε > 0 such that if ϕ is an eigenfunction of T2 of eigenvalue E = λ2 and C any regular arc contained in Nϕ , then w(C) < cε λ−1+ε .

(1.7)

This is our substitute for the phenomenon M. Berry pointed out for random plane waves. The above conjecture seems to be consistent with numerics and we will moreover prove its validity for ‘most’ eigenvalues E. In generality, we prove Theorem 1. If C ⊂ Nϕ is a regular arc, then (1.8)

1

w(C) < Cε λ− 3 +ε .

The argument makes crucial use of the structure of lattice points on the circle {|ξ| = λ}. Relevant results will be presented in § 2. In § 6 we will show that the exponent 1/3 of Theorem 1 can be improved to 1/2 for almost all of the nodal line, in the following sense: Theorem 2. Given ε > 0, there is δ > 0 such that the following holds. Let {Cα } be a collection of disjoint regular arcs of Nϕ satisfying (1.9) Then for λ > λ(ǫ), (1.10)

1

w(Cα ) > λ− 2 +ε for each α. X

length(Cα ) < λ1−δ

α

Recall that length(Nϕ ) ≈ λ, by (1.3), so that Theorem 2 asserts that arcs of large width form a negligible part of the nodal set. As we will see, the exponent 21 in (1.9) could be replaced by 1, assuming the validity of the Cilleruelo-Granville conjecture [C-G], stating that for all


4

ε > 0, there is a constant Bε such that any arc on a circle {|ξ| = λ} of size at most λ1−ε contains at most Bε lattice points (uniformly in λ). Finally, we also show that Conjecture 1 holds for at least a positive proportion of the nodal set, in the following sense: Theorem 3. There is a constant c0 > 0 such that the following holds. Let ε > 0, λ large enough and {Cα } a collection of disjoint regular arcs of Nϕ satisfying w(Cα ) > λ−1+ε for each α.

(1.11) Then

length(Nϕ \

(1.12)

[

Cα ) > c0 λ.

α

The proofs of Theorems 2 and 3 make essential use of the results of Donnelly and Fefferman [D-F]. 1.2. Total curvature. Another geometric characteristic of nodal lines that one can investigate is their total curvature. For C 2 curves in R3 , if γ : [0, ℓ] → C is a C 2 arc length parametrization then the total curvature is Z ℓ ||¨ γ (s)||ds . (1.13) K(C) = 0

When one varies the curve C, the formula (1.13) is clearly continuous in the C 2 topology and hence can be used to define the total curvature of any continuous curve as the limit of the total curvature of its smooth perturbations. However, there is definition of total curvature which makes sense for any continuous curve, which starts with defining the total curvature of a polygon as the sum of the angles subtended by the prolongation of any of its sides and the next one, and then for any continuous curve C setting (1.14)

K(C) = sup K(P ) P 1

where the supremum is over all polygons P inscribed in C. One can show that for C 2 curves this definition coincides with (1.13) (see [M]). We claim that the total curvature Kϕ of the nodal set for an eigenfunction ϕ with eigenvalue E is bounded by (1.16)

Kϕ ≪ E

Note that there is no lower bound, since the nodal set of the eigenfunction sin nx is a union of non-intersecting lines hence has zero total curvature. 1Alternatively one can take

(1.15)

K(C) = lim K(P ) P

where the limit is over all polygons P inscribed in C for which the maximal distance between adjacent vertices tends to zero; this definition works for curves in arbitrary Riemannian manifolds [CL].


5

To prove (1.16), it suffices to assume that nodal set is smooth, which is easily seen to be a generic condition in the eigenspace on the torus, hence a small perturbation in the eigenspace will bring us to that setting and one then invokes continuity of the total curvature in the C 2 -topology. In case the nodal set is smooth, one can make the following comment based on the fact that Nϕ is a semi-algebraic set. First observe that ϕ in (1.1) may be expressed as X (1.17) ϕ(x1 , x2 ) = aα,β (cos x1 )α1 (cos x2 )α2 (sin x1 )β1 (sin x2 )β2 α,β

√ with α, β ∈ Z2+ and α1 + α2 + β1 + β2 ≤ 2λ. Introducing variables u1 = cos x1 , u2 = cos x2 , it follows that u1 , u2 satisfy a polynomial equation (1.18)

P (u1 , u2 ) = 0

with P ∈ R[u1 , u2 ] of degree d < cλ. According to [R, Theorem 4.1, Proposition 4.2], assuming {P = 0} is smooth, its total curvature2 is at most const.d2 ≪ E. Since Nϕ ⊂ {P = 0} (in the u1 , u2 -parametrization), we may conclude that (1.16) holds. 1.3. Remarks. (1) In defining regular arcs, one could make further higher derivative assumptions on the parametrization γ (as we will show with an example in Appendix A, those do not hold automatically). Involving higher derivatives would allow to improve upon the estimate (1.8). We do not pursue this direction here however partly because Definition 1 would have to be replaced by a more technical one and it is not clear which version would be the most natural. (2) We point out that our estimates for the width are specific to the flat torus. For instance, they are not valid on the sphere S 2 . Indeed, the standard spherical harmonics Yℓ,m = Pℓ,m (θ)eimϕ are eigenfunctions for which the circles of latitude {Pℓ,m (θ) = 0} are families of regular arcs with geodesic curvature bounded away from zero. (3) One can easily obtain the analogue of (1.16) for the total curvature of the nodal sets on the sphere using similar arguments to those on the torus. At the time of this writing, it is not clear to us if there is an estimate of the type (1.16) for general real-analytic surfaces, or even, more modestly, any explicit bound Kϕ < K(E) for the total curvature. 2Since the total curvature of an arc is the variation of its tangent vector, a bound is

obtained by integration in s of the number of solutions in u = (u1 , u2 ) of ( ∂2 P (u) + s∂1 P (u) = 0 P (u) = 0.

6


Acknowledgements: We thank Misha Sodin for his comments. J.B. was supported in part by N.S.F. grant DMS 0808042. Z.R. was supported by the Oswald Veblen Fund during his stay at the Institute for Advanced Study and by the Israel Science Foundation (grant No. 1083/10). 2. Lattice points on circles In this section, we collect some facts about lattice points on arcs for later use. Let E = λ2 ∈ Z+ and E = Z2 ∩ {|x| = λ}

Then |E| = r2 (E) is the number of representations of E as a sum of two squares, which is essentially the number of divisors of E in the ring of Gaussian integers. In particular one has an upper bound log E ≪ E ε for all ε > 0 . (2.1) r2 (E) ≪ exp c log log E The next statement is a slight specification of a more general result due to Jarnik [J]. Lemma 1. Let P0 , P1 , P2 ∈ E be distinct and |P0 − P1 | ≤ |P0 − P2 |. Then (2.2)

|P0 − P2 |2 .|P0 − P1 | > cλ

(here and in the sequel, c, C will denote constants). Proof. P0 , P1 , P2 belong √ to an arc C ⊂ {|x| = λ} of size r and we may obviously assume r < λ. Since P0 , P1 , P2 are distinct, they span a triangle T of area   1 P0 1 1 0 < area(T ) = det 1 P1  ∈ Z+ . 2 2 1 P 2

Hence, from geometric considerations

1 r2 ≤ area (T ) < c .|P0 − P1 | 2 λ Lemma 2. Let P0 , P1 , Q0 , Q1 ∈ E be distinct points on an arc of size r. Then (2.3)

|P0 − Q0 |.|P1 − Q1 |.r > cλ.

Proof. We may assume r
g(x0 ) = f (x0 ), while if x0 < t < L then g′ (t) = −tf ′′ (t) > 0 so g is increasing and so g(t) > g(x0 ) = f (x0 ). Having established that w(C) ≈ f (x0 ), it remains to show that f (x0 ) ≈ κℓ2

Assuming say that x0 ≤ L/2, we expand f in a Taylor series around the endpoint x = L further from x0 , finding 1 0 = f (L) = f (x0 ) + f ′ (x0 )(L − x0 ) + f ′′ (y)(L − x0 )2 2 for some x0 < y < L. Using f (L) = 0, f ′ (x0 ) = 0 and f ′′ < 0, |f ′′ | ≈ κ and L/2 ≤ L − x0 ≤ L we get (3.6)

1 f (x0 ) = − f ′′ (y)(L − x0 )2 ≈ κL2 2 Now note that L ≈ ℓ because Z Lp ℓ= 1 + f ′ (t)2 dt ∈ [L, 3L] 0

using

|f ′ (x)|

< 2. Hence f (x0 ) ≈ κℓ2 as claimed.

4. Local estimates on the width 4.1. Fourier transforms of arcs. We establish some bounds on the Fourier transform of measures supported by “regular” arcs. Let γ : [0, ℓ] → C be an arc-length parameterization of the regular arc .. C, so that |γ| ˙ = 1, and κ < |γ| < 2κ. Note that if ξ ∈ R2 , |ξ| = 1 and ℓκ 0 < ρ < 10 , then (4.1)

Iξ = {t ∈ I : |ξ · γ(t)| ˙ < ρ}

is an interval of size at most O(ρ/κ).

12


Indeed, the length of Iξ can be computed as Z Z du dt = (4.2) ¨ (t)| |u| κ 1 − ρ2 and hence (4.4) since ρ < ℓκ/10 < 1/10.

ρ 2ρ ≪ length Iξ < p 2 κ κ 1−ρ

Lemma 7. Let ξ ∈ R2 \{0} and assume ξ ˙ > ρ for all t ∈ [0, ℓ] . (4.5) · γ(t) |ξ|

Let ω : R → R+ , supp ω ⊂ [0, ℓ] satisfy Z Z c (4.6) ω = 1 and |ω ′ | < . ℓ Then Z c 1 κ (4.7) + e ξ · γ(t) ω(t)dt < ρ|ξ| ℓ ρ (where c denotes various constants).

Proof. A change of variables u = ξ · γ(t) gives Z n .. Z d h ω(t) i Z |ω(t)| ˙ ω(t)|ξ · γ(t)| o + e ξ · γ(t) ω(t)dt ≤ dt ≤ 2 dt ξ · γ(t) ˙ |ξ · γ(t)| ˙ |ξ · γ(t)| ˙ Z Z κ κ c 1 + 2 |ω(t)| ˙ + 2 ω(t) < ≤ ρ|ξ| ρ |ξ| ρℓ|ξ| ρ |ξ| from the assumptions. √ 1 Fix E ∈ Z+ (large), λ = E and let E = Z2 ∩ {|x| = λ}. Fix 0 < c0 < 100 ξ −ξ and take ρ = c0 κℓ/|E|2 . We let ξ run over all vectors ξ = |ξ11 −ξ22 | , ξ1 6= ξ2 in E. Excluding the corresponding subintervals Iξ of (4.1) from I, of length |Iξ | < c0 ℓ|E|−2 , we obtain Lemma 8. There is a collection of at most |E|2 disjoint sub-intervals Iτ ⊂ I with the following properties: (4.8) (4.9)

|Iτ | > c0 |E|−2 ℓ X |Iτ | > (1 − 2c0 )ℓ


(4.10)

κℓ ξ1 − ξ2 ˙ · > c0 2 γ(t) |ξ1 − ξ2 | |E|

13

for ξ1 6= ξ2 in E and t ∈ I

Let ω : R → R+ , supp ω ⊂ Iτ satisfy Z Z |E|2 (4.11) ω = 1 and |ω| ˙ . . c0 ℓ Then for all ξ1 6= ξ2 in E Z (4.12) e (ξ1 − ξ2 ) · γ(t) ω(t)dt .

where ω(C) = ℓ2 κ is the width of C.

1 |E|4 2 c0 ω(C) |ξ1 − ξ2 |

The estimate (4.12) follows indeed from (4.7) and the above choice of ρ. Returning to Theorem 1, we simply replace I by some Iτ and C by Cτ = γ(Iτ ). Redefining ℓ = |Iτ |, we have for all ξ1 6= ξ2 ∈ E the estimate Z λε (4.13) e (ξ1 − ξ2 ) · γ(t) ω(t)dt ≪ ω(C)|ξ1 − ξ2 |

if ω : R → R+ , supp ω ⊂ I satisfies Z (4.14) ω = 1,

Z

|ω| ˙ .

1 . ℓ

4.2. The exponent 1/6. As a warm-up, we show how to prove Conjecture 1 for almost all energies E and how to obtain a weaker version of Theorem 1 with the exponent 1/6 instead of 1/3. Consider the Fourier expansion of ϕ: X (4.15) ϕ(x) = ϕ(ξ)e(x b · ξ). ξ∈E

P 2 = ||ϕ||2 , we have Since the Fourier coefficients of ϕ satisfy b 2 ξ∈E |ϕ(ξ)| |ϕ(ξ)| b ≤ ||ϕ||2 for all ξ ∈ E and hence there is some ξ0 for which ||ϕ||2 |ϕ(ξ b 0 )| ≥ p |E|

Replacing ϕ by ϕ/ϕ(ξ b 0 ), we may thus assume p (4.16) ϕ(ξ b 0 ) = 1, ||ϕ||2 ≤ |E|

and in particular |ϕ(ξ)| b ≪ λǫ for all ǫ > 0. Assume C ⊂ Nϕ . Since ϕ γ(t) = 0, we obtain for any weight function ω as in Lemma 7 that Z 0 = ϕ γ(t) e − ξ0 · γ(t) ω(t)dt I Z (4.17) X =1+ e (ξ − ξ0 ) · γ(t) ω(t)dt ϕ(ξ) b ξ6=ξ0

I


14

By Jarnik, there is at most one frequency ξ1 6= ξ0 at distance ≪ λ1/3 from ξ0 . P For all other frequencies we use (4.12) together with ξ |ϕ(ξ)| b 2 ≤ |E| ≪ λǫ to get Z b λ−1/3+ǫ λǫ X |ϕ(ξ)| ≪ (4.18) 1 + ϕ(ξ b 1 ) e (ξ1 − ξ0 ) · γ(t) ω(t)dt ≪ w |ξ − ξ0 | w ξ6=ξ0 ,ξ1

We may now show that Conjecture 1 holds for almost all E. First choose E = λ2 satisfying (2.10). Then ξ1 does not exist and |ξ − ξ0 | > λ1−ǫ for ξ 6= ξ0 , hence (4.18) gives 1 ≪ λ−1+ǫ /w, that is w ≪ λ−1+ǫ . Returning to the case of general E, if there is no such ξ1 , that is if ξ0 is at distance at least λ1/3 from all other frequencies, then (4.18) implies w ≪ λ−1/3+ǫ . Otherwise, that is if there is a neighbour ξ1 , we proceed as follows: Start by performing a rotation T of the plane as to insure T (ξ1 − ξ0 ) = (|ξ1 − ξ0 |, 0) ∈ R2 .

(4.19)

Denoting T γ again by γ = (γ1 , γ2 ), we obtain from (4.18) that Z λ−1/3+ǫ (4.20) 1 + ϕ(ξ b 1 ) e (|ξ1 − ξ0 |γ1 (t) ω(t)dt ≪ w ˙ Next we specify ω. Writing γ(s) ˙ = eiθ(s) , we have θ(s) ∼ κ (or −κ, which is similar) and p .. ˙ γ 1 (s) = − sin θ(s) θ(s) ∼ − 1 − γ˙ 1 (s)2 κ.

Therefore there is a suitable restriction of s ∈ I1 ⊂ I, |I1 | ∼ ℓ and some ρ & κℓ (recall that κℓ < 1) such that ρ (4.21) |γ˙ 1 (s) − ρ| < cκℓ < for s ∈ I1 . 10 Let s0 ∈ I1 be the center of I1 . Define γ˙ 1 (s)η γ1 (s) − γ1 (s0 ) (4.22) ω(s) = R ˙ 1 (s)η γ1 (s) − γ1 (s0 ) ds I1 γ R x where η is a bump-function of the form η(x) = η0 ( ρℓ ) with η0 ≥ 0, η0 = 1 (see Figure 4), chosen as to ensure that supp ω ⊂ I1 (we use (4.21) here). 0.8 0.6 0.4 0.2

-1.0

-0.5

0.5

Figure 4. The bump function η0 (t)

1.0


Also

15

Z c ω=1 0≤ω< , ℓ Z 1 κ 1 |ω ′ | . + . ρ ℓ ℓ

and

and (4.14) holds. With the choice (4.22) and change of variable u = γ1 (s) − γ1 (s0 ), one obtains in (4.38) (4.23) R Z e(|ξ1 − ξ0 |u)η(u)du R e |ξ1 − ξ0 |γ1 (t) ω(t)dt = e |ξ1 − ξ0 |γ1 (s0 ) η(u)du Z 1 = e |ξ1 − ξ0 |γ1 (s0 ) e(|ξ1 − ξ0 |u)η(u)du ρℓ Z = e |ξ1 − ξ0 |γ1 (s0 ) e(ρℓ|ξ1 − ξ0 |t)η0 (t)dt since η0 (t) = η(ρℓt). Thus we find

λ−1/3+ǫ 1 + e |ξ1 − ξ0 |γ1 (s0 ) b ≪ w

(4.24) where

b = ηb0 (|ξ1 − ξ0 |ρℓ)ϕ(ξ b 1)

(4.25)

satisfies |b| ≪ λǫ . Note that our choice of s0 ∈ I1 allows moving s0 within an interval I2 ⊂ I1 of size |I2 | = 21 |I1 | ∼ ℓ. Since γ1 (I2 ) contains an interval of size at least ∼ ρℓ & w, it follows that (4.26) max 1 + be |ξ1 − ξ0 |γ1 (s0 ) ≥ max 1 + be(u) s0 ∈I2

u∈U

where U ⊂ R is some interval of size ∼ |ξ1 − ξ0 |w. Then we have 1 (4.27) (4.26) ≥ min(1, w|ξ1 − ξ0 |) 2 Indeed, if |b| ≥ 3/2 or |b| ≤ 1/2 then |1 + be(u)| ≥ 1/2, while if 1/2 < |b| < 3/2 then we can bound 1 max |1 + be(u)| ≥ |b| max | sin u| ≥ |U | u∈U u∈U 4 Thus we find λ−1/3+ǫ (4.28) min(1, w|ξ1 − ξ0 |) ≪ w −1/3+ǫ If the minimum is 1, we get w ≪ λ . Otherwise (taking into account |ξ1 − ξ0 | ≥ 1) we get w ≪ λ−1/6+ǫ


16

4.3. Proof of Theorem 1. Fix some ξ0 ∈ E and enumerate E = ξ0 , ξ1 , . . . such that |ξ0 − ξj | ≤ |ξ0 − ξj+1 |.

(4.29) Write (4.30)

ϕ(x) =

X j

Let 1 < r
J

Perform a rotation T of the plane as to insure (4.34)

T (ξJ − ξ0 ) = (ξJ′ , 0),

ξJ′ = |ξJ − ξ0 |

and denote (4.35)

T (ξj − ξ0 ) = (ξj′ , ζj′ ) ∈ R2 .

Clearly (4.36)

ξj′ ∼ |ξj − ξ0 |,

j≤J

and r |ζj′ | < 2 |ξj − ξ0 |. λ Denoting T γ again by γ = (γ1 , γ2 ), we easily obtain i X hZ (4.32) = c0 + (4.38) e ξj′ γ1 (t) ω(t)dt cj (4.37)

1≤j≤J

(4.39)

r X + 0 |cj | |ξj − ξ0 | . λ 0 (1 − 2c0 )|Cα | τ

Z (6.4)

Cα,τ

ds |E|4 |ξ1 − ξ2 |−1 ≪ λρ+ε |ξ1 − ξ2 |−1 e (ξ1 − ξ2 ) · x . 2 |Cα,τ | c0 w(Cα )

for all ξ1 6= ξ2 . Here ds stands for the arc-length measure on Cα ; ε > 0 is arbitrarily small. We get a subset N1 ⊂ N0 defined by [[ (6.5) N1 := Cα,τ . α

τ

Using (6.3) and the Donnelly-Fefferman upper bound (1.3) we see that X |Cα | ≤ 2c0 length(Nϕ ) . c0 λ (6.6) length(N0 \N1 ) < 2c0 α

Fix ρ < ρ1 = ρ + 3δ < 1 and introduce a partition [ (6.7) E= Eβ β

of the lattice points E = Z2 ∩ {|ξ| = λ}, satisfying (6.8)

dist(Eβ , Eβ ′ ) > λρ1 for β 6= β ′

(6.9)

diam Eβ ≪ λρ1 +ε for each β.

The construction is straightforward: If we introduce a graph on E, defining ξ ∼ ξ ′ if |ξ−ξ ′ | ≤ λρ1 , its connected components Eβ are obviously of diameter at most λρ1 · #E ≪ λρ1 +ε and (6.8) holds. Let

ϕ=

X

ϕβ ,

kϕk2 = 1

be the corresponding decomposition of our eigenfunction ϕ. Thus supp ϕ bβ ⊂ Eβ . For each α, τ we have by (6.4) Z XZ 2 |ϕ(x)| = (6.10) 0 = |ϕβ |2 Cα,τ

Cα,τ

β

+O

X X

β6=β ′ ξ,∈Eβ ξ ′ ∈Eβ ′

|ϕ bβ (ξ)||ϕ bβ ′ (ξ ′ )|

λρ+ε |Cα,τ | |ξ − ξ ′ |

and (6.8) implies the bound λρ−ρ1 +ε |Cα,τ | on the last term of (6.10). Summing (6.10) over all α, τ gives XZ (6.11) |ϕβ |2 ≪ λ1+ρ−ρ1 +ε . β

N1


24

p P Since β kϕβ k22 = 1, one can specify some β such that ||ϕβ || ≥ 1/ |E| and hence ϕβ ψ := kϕβ k2 has kψk2 = 1 and satisfies Z |ψ|2 ≪ λ1+ρ−ρ1 +ε . (6.12) N1

and

ψ=

X

ξ∈E ′

E′

b ψ(ξ)e(x · ξ)

with = Eβ ⊂ E contained in an arc of size r < λρ1 +ǫ . Now define (6.13) (recall δ =

N ′ := {x ∈ N , |ψ(x)| < λ−δ },

ρ1 −ρ 3 ).

N1′ := N1 ∩ N ′

It follows from (6.12) that

length(N1 \N1′ ) < λ1−δ+ǫ .

(6.14)

Consider a partition of T2 in squares Qν of size [ Ω= Qν ⊂ T 2 .

1 λ

and let

Qν ∩N ′ 6=φ

We wish to bound the area |Ω|.

First, observe that in general for x, y ∈ T2 b 1 [diam(supp ψ)]|x b ψ(x) − ψ(y) ≤ kψk − y|

and hence (6.15)

1

sup ||ψ(x)| − |ψ(y)|| . |E ′ | 2 rλ−1 < λ−

1−ρ1 2

.

x,y∈Qν

It follows that (6.16)

sup |ψ| < λ−δ + λ−

1−ρ1 2

Ω

6.1. Proof of Theorem 2 (ρ < 21 ). Let ρ = δ > 0, so that (6.17)

1

r < λ 2 −δ

and sup |ψ| < λ−δ Ω

From (5.8) and the preceding λ−δ > (c|Ω|)1/δ implying (6.18)

2

|Ω| < λ−δ .

1 2

− 5δ, ρ1 =

1 2

− 2δ for some


25

2

Thus Ω contains at most λ2−δ boxes Qν , and Lemma 13 implies X δ2 2 (6.19) length(N ′ ) = length(N ∩ Qν ) ≪ λ1−δ +ε < λ1− 2 Qν ∩N ′ 6=∅

for λ large enough. Thus

δ2

length(N1′ ) < λ1− 2 .

(6.20)

Using (6.14), we therefore get δ

(6.21)

length(N1 ) < λ1− 2

and since length(N0 \N1 )
c1 λ Qν ∩Ω=φ

where c1 is some absolute constant. Hence, from (6.6), (6.14), (6.25)

length(Nϕ \N0 ) ≥ length(Nϕ \N1′ ) − length(N0 \N1 ) − length(N1 \N1′ ) 1 > c1 λ − c0 λ − λ1−δ+ǫ > c1 λ 2 if we choose c0 small enough. This proves Theorem 3.

Appendix A. Higher order regularity - An example The purpose of what follows is to show that ‘regular arcs’ need not satisfy higher order smoothness bounds, even for κ small. Consider the eigenfunction (A.1)

ϕ(x, y) = sin(ky + x) + ε sin(ky − x) + δ sin(y + kx)

with eigenvalue E = 1 + k2 , where

ε = 10−10 and δ = 10−100 k−2 .


26

We restrict x ∈ I = − π4 + [−10−3 , 10−3 ] and consider the curve C ⊂ {ϕ = 0} parameterized by y = y(x), x ∈ I, such that (A.2)

Evaluate

y ′ , y ′′ , y ′′′

|ky(x) + x| = 0(ε).

cos(ky + x)(ky ′ + 1) + ε cos(ky − x)(ky ′ − 1) + δ cos(y + kx)(y ′ + k) = 0 (A.3) y ′ =

cos(ky + x) − ε cos(ky − x) + δk cos(y + kx) 1 ε = − +0 +δ k[cos(ky + x) + ε cos(ky − x)] + δ cos(y + kx) k k

and [k cos(ky + x) + kε cos(ky − x) + δ cos(y + kx)]y ′′ =

sin(ky + x)(ky ′ + 1)2 + ε sin(ky − x)(ky ′ − 1)2 + δ sin(y + kx)(y ′ + k)2 =

(since ϕ = 0)

ε sin(ky − x)[(ky ′ − 1)2 − (ky ′ + 1)2 ] + δ sin(y + kx)[(y ′ + k)2 − (ky ′ + 1)2 ] = −4kε sin(ky − x)y ′ − δ(k2 − 1) sin(y + kx) (y ′ )2 − 1

and

′′

y =

+ 0( kε ) + 0(δk 2 ) k 1 + 0(ε + kδ )

−4kε sin 2x + 0(ε)

=−

1 k

ε2 4ε sin 2x + 0 + δk k k

4ε k from the choice of ε, δ and I. Thus C is convex with curvature ∼ k1 . Next, from the preceding k 1 + 0(ε) y ′′′ + 0(kε|y ′′ |) = 0(kε|y ′ | + kε|y ′′ | + k2 δ|y ′ | |y ′′ |)+ δ(k2 − 1) 1 − (y ′ )2 (k + y ′ ) cos(y + kx) ∼

and

ε 1 1 + 0(ε) y ′′′ = δ(k2 − 1) cos(y + kx) + 0 + δ = δk2 cos kx + 0 k k 2 −100 (iv) where δk = 10 . It follows that ky k∞ ∼ k for large k. References

[Be] M. Berry Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J. Phys. A 35 (2002), no 13, 3025–3038. [CL] M. Castrill´ on L´ opez, V. Fern´ andez Mateos and J. Mu˜ noz Masqu´e. Total curvature of curves in Riemannian manifolds. Differential Geom. Appl. 28 (2010), no. 2, 140-147. [Ch] S. Y. Cheng Eigenfunctions and nodal sets. Comment. Math. Helv. 51 (1976), no. 1, 43-55.


27

[C-C] J. Cilleruelo, A. Cordoba Trigonometric polynomials and lattice points, Proc. AMS, 115 (4) (1992), 899–905. [C-G] J. Cilleruelo, A. Granville Lattice points on circles, squares in arithmetic progressions and sumsets of squares, CRM Proc. LN, Vol. 43, AMS 2007, 241–262. [C-G2] J. Cilleruelo, A. Granville Close lattice points on circles, Canad. J. Math. 61 (2009), no 6, 1214–1238. [D-F] H. Donnelly, C. Fefferman Nodal sets of eigenfunctions of Riemannian manifolds, Inventiones math. 93 (1988), no 1, 161–183. [F-M] N. Fontes-Merz A multidimensional version of Tur´ an’s lemma. J. Approx. Theory 140 (2006), no. 1, 27-30. [J] V. Jarnik Uber die Gitterpunkte auf konvexen Kurven, Math. Z. 24 (1) (1926), 500– 518. [M] J. W. Milnor, On the total curvature of knots. Ann. of Math. (2) 52, (1950). 248-257. [N] F. Nazarov Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5 (1993), no. 4, 3–66; translation in St. Petersburg Math. J. 5 (1994), no. 4, 663-717. [Ra] D. S. Ramana Arithmetical applications of an identity for the Vandermonde determinant. Acta Arith. 130 (2007), no. 4, 351–359. [R] J. Risler On the curvature of the real Milnor fiber, Bull. London Math. Soc. 35 (2003), 445–454. School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 E-mail address: [email protected] Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail address: [email protected]