Nodal intersections for random waves on the 3-dimensional torus

0.27 MB·English

by  Zeev Rudnick

NODAL INTERSECTIONS FOR RANDOM WAVES ON
THE 3-DIMENSIONAL TORUS
6
1
0
Z. RUDNICK,I.WIGMAN AND N.YESHA
2

y
Abstract. Weinvestigatethenumberofnodalintersectionsofrandom
a
M Gaussian Laplaceeigenfunctionsonthestandardthree-dimensionalflat
toruswithafixedsmoothreferencecurve,whichhasnowherevanishing

2 curvature. Theexpectedintersectionnumberisuniversallyproportional
1 tothelengthofthereferencecurve,timesthewavenumber,independent
ofthegeometry. Ourmainresultgivesaboundforthevariance,ifeither

] the torsion of thecurveis nowhere zero or if thecurveis planar.
T
N
.
h
t 1. Introduction
a
m
1.1. Toral nodal intersections. Let Td = Rd/Zd be the standard flat d-
[ dimensional torus and Td a fixed reference curve1. Given a real-valued

C ⊂
3 eigenfunction F(x) of the Laplacian
v
0 ∆F = 4π2E F ,
− ·
1
we wish to study the number of intersections
4
7 (F) := # x Td :F(x) = 0
0 Z { ∈ }∩C
1. of the nodal set of F with the reference curve as a function of the corre-
C
0 sponding eigenvalue.
5
Indimensiond = 2, ifthecurveissmoothandhasnowherevanishingcur-
1
vature, then deterministically for every eigenfunction, the number of nodal
:
v
intersections satisfies [3, 4]
i
X
E1/2
r (F) E1/2;
a (logE)5/2 ≪ Z ≪
hereand everywheref g (equivalently f = O(g)) means that there exists
≪
a constant C > 0 such that f C g . For the upper bound we require
| | ≤ | |
that is real-analytic, in particular it is shown that is not contained in
C C
the nodal set for E sufficiently large. One can improve the lower bound to
(F) E1/2 conditionally on a certain number-theoretic conjecture [4].
Z ≫
Date: May 13, 2016.
TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanResearch
CouncilundertheEuropean Union’sSeventhFramework Programme (FP7/2007-2013) /
ERC grant agreements no 320755 (Z.R.) and no 335141 (I.W.), and from the Friends of
theInstitutefor Advanced Study(Z.R.).
1By a curvewe always mean a parameterized, compact, immersed curve.
1
2 Z.RUDNICK,I.WIGMANANDN.YESHA
In this note we deal with dimension d = 3. If we consider the intersection
of the nodal set with a fixed real-analytic reference surface Σ T3 with
⊂
nowhere-zero Gauss-Kronecker curvature, then for E sufficiently large, Σ is
not contained in the nodal set [2], the length of the intersection of Σ with
the nodal set is √E [3], and the nodal intersection is non-empty [3].
≪
However, for the intersection of the nodal set with a fixed reference curve
(whereweexpectonlyfinitelymany points),thefollowing examplesindicate
that one cannot expect to have any deterministic bounds on the number of
nodal intersections.
Example 1.1. Take the eigenfunctions of the form
F (x ,x ,x )= sin(2πkx );
k 1 2 3 1
their nodal surfaces are the planes x T3 : x 1 Z and any curve lying
∈ 1 ∈ 2k
on the plane x = 1/2π does not intersect these, whereas a curve on the
1
(cid:8) (cid:9)
plane x = 0 is lying inside all of these nodal surfaces. We observe that the
1
curves in this example are planar.
Example 1.2. Let F (x,y) be an eigenfunction on the two-dimensional
0
toruswitheigenvalue4π2E2,andS acurvedsegmentcontainedinthenodal
0 0
set, admitting an arc-length parameterization γ : [0,L] S , with curva-
0 0
→
ture κ (t) = γ (t) > 0. For n 0 let F (x,y,z) = F (x,y)cos(2πnz), an
0 | 0′′ | ≥ n 0
eigenfunction on T3 with eigenvalue 4π2(E2+n2). Let be the parametric
0 C
curve γ(t) = γ ( t ), t . A computation shows that the curvature is
0 √2 √2
κ(t) = 1κ ( t )(cid:16)> 0, and th(cid:17)at the torsion is τ(t)= 1κ ( t )= 0, so that
2 0 √2 ±2 0 √2 6 C
is non-planar. Clearly is contained in the nodal set of F for all n. Thus
n
C
even in the non-planar case, we can have the reference curve contained in
C
the nodal set for arbitrarily large E.
Question 1.3. Does there exist a non-planar curve with no nodal inter-
C
sections, E arbitrarily large?
1.2. Arithmetic random waves. As there is no deterministic bound on
(F) in dimension 3, we investigate what happens for “typical” eigenfunc-
Z
tions. Let (E) be the set of lattice points lying on a sphere
E
(E) = ~x Z3 : ~x 2 = E ,
E { ∈ | | }
and let
N = N := # (E).
E
E
The Laplace spectrum on T3 is of high multiplicities, with the dimension
of an eigenspace corresponding to an eigenvalue 4π2E being of size N
E
≈
E1/2 o(1). The eigenfunctions corresponding to the eigenvalue 4π2E are of
±
the form
1
(1.1) F(x) = F (x) = a e2πi µ,x .
E µ h i
√N
E
µ (E)
∈XE
NODAL INTERSECTIONS FOR RANDOM WAVES ON THE 3-DIM TORUS 3
We consider random Gaussian eigenfunctions (“arithmetic random waves”
[11]) by taking the coefficients a to be standard complex Gaussian random
µ
variables, independent save for the relations a = a , making F real-
µ µ
−
valued.
1.3. Statement of the main results.
Theorem 1.4. Let T3 be a smooth curve of length L, with nowhere
C ⊂
zero curvature. Assume further that one of the following holds:
(i) has nowhere-vanishing torsion;
C
(ii) is planar (so that the torsion vanishes identically).
C
Then for all ǫ > 0, as E along integers E 0,4,7 mod 8, the number
→ ∞ 6≡
of nodal intersections satisfies
(F) 2
lim Prob Z L > ǫ 0.
E √E − √3 →
E 0,4→,7∞mod8 (cid:18)(cid:12) (cid:12) (cid:19)
6≡ (cid:12) (cid:12)
(cid:12) (cid:12)
Note that the condition E 0(cid:12),4,7 mod8 is (cid:12)natural, as otherwise E =
6≡
4aE with E 7 mod8 (if E 7 mod8 then E is not a sum of three
′ ′ ′
6≡ ≡
squares and hence does not yield a Laplace eigenvalue); then an eigenfunc-
tion of eigenvalue 4π2E is necessarily of the form F(x) = H(2ax) with H
an eigenfunction of eigenvalue E . Hence any question on the nodal set of
′
F reduces to the corresponding question on the nodal set of eigenfunctions
with eigenvalue E = E/4a (which may be trivial, e.g. if E = 4a).
′
To prove Theorem 1.4 we compute the expected value of with respect
Z
to the Gaussian measure defined on the eigenspace as above to be
2
E( ) = L√E,
Z √3
and give an upper bound for the variance:
Theorem 1.5. Let T3 be a smooth curve, with nowhere-zero curvature.
C ⊂
Assume also that either has nowhere-vanishing torsion, or is planar.
C C
Then for E 0,4,7 mod8,
6≡
1
(1.2) Var Z
√E ≪ Eδ
(cid:18) (cid:19)
for all δ < 1/3 in case has nowhere vanishing torsion, and all δ < 1/4 in
C
case is planar.
C
Remark 1.6. If is real-analytic and non-planar, so that the torsion is not
C
identically zero, but may vanish at finitely many points, the result (1.2)
above is valid with some δ = δ > 0, see 5.3.
C §
1.4. Outlineof thepaper andthekeyideas. Weprovetheapproximate
Kac-Rice formula (briefly explained in 1.5) in 2, followed by a study of
§ §
certain oscillatory integrals on the curve in 3. The arithmetic heart of the
§
paper is 5, where we bound the second moment of the covariance function
§
4 Z.RUDNICK,I.WIGMANANDN.YESHA
and its derivatives, following some background on the arithmetic of sums of
three squares in 4, expanded on in Appendix A.
§
AsimilarresulttoTheorem1.5wasprovedinthetwo-dimensionalcasefor
T2 having nowhere-zero curvature [13]. In that case the authors found
C ⊂
that the precise asymptotic behaviour of the nodal intersections variance
is non-universal, namely dependent on both the angular distribution of the
latticepoints (E)andthegeometryof . Inthe3-dimensionalcasewewere
E C
only able to obtain an upper bound (1.2) on the variance, which implies the
“almost-all” statement of Theorem 1.4. Thesetwo cases differboth interms
of analytic and arithmetic ingredients; the arithmetic of ternary quadratic
forms differs significantly from that of binary quadratic forms.
1.5. An approximate Kac-Rice formula. WeendthisIntroductionwith
a discussion of a key step in our work, the approximate Kac-Rice formula.
By restricting the arithmetic random waves (1.1) along , the problem of
C
nodal intersections count is reduced to evaluating the number of zeros (zero
crossings) of a random (nonstationary) Gaussian process. The Kac-Rice
formula is a standard tool or meta-theorem for expressing the (factorial)
moments of the zero crossings number of a Gaussian process precisely in
terms of certain explicit integrals with integrands depending on the given
process. It is then easy to evaluate the expected number of zeros precisely
and explicitly via an evaluation of a standard Gaussian expectation.
For the variance, or the second factorial moment, the validity of the Kac-
Rice formula is a very subtle question with a variety of sufficient conditions
known in the literature (e.g. [8, 1]). While the classical treatise [8] re-
quiresthe non-degeneracy of the(Gaussian) distribution of the values of the
given process at two points together with their derivatives, only the non-
degeneracy ofthevalues distribution(at twopoints)isrequiredforthemore
modern treatment [1], which, to the best knowledge of the authors, is the
weakest known sufficient condition for the validity of Kac-Rice. Unfortu-
nately, even this weaker condition may fail in our case.
In orderto treat this situation ( 2)we dividetheinterval into many small
§
subintervals of length commensurable to the wavelength 1 and decompose
√E
the total variance as a sum over pairs of subintervals of zero number co-
variances (2.14). We were able to prove the validity of Kac-Rice for most
of the pairs of subintervals that includes all the diagonal pairs (i.e. the
variance of nodal intersections along sufficiently small curves), and bound
the contribution of the other pairs via a simple Cauchy-Schwartz argument.
The price is that along the way we incur an error term, hence yielding an
approximate Kac-Rice (Proposition 2.2) reducing a variance computation to
anestimate forsomemoments of thecovariance functionand its derivatives;
such a strategy was also used in [13, 7]. The remaining part of this paper
is concerned with proving such an estimate on the second moment of the
covariance function and a couple of its derivatives.
NODAL INTERSECTIONS FOR RANDOM WAVES ON THE 3-DIM TORUS 5
We finally record that we may omit a certain technical assumption made
in our previous work on the two-dimensional case [13] by using the more
general form of the Kac-Rice formula as in [1] (vs. [8]). Given an integer m
expressible as a sum of 2 squares we defined the probability measure
τ = δ
m λ/√m
λ 2=m
k Xk
on the unit circle 1 R2 supported on all the lattice points λ Z2 lying
S ⊆ ∈
on the centered radius-√m circle in R2 projected to the unit circle. In [13,
Theorem1.2] weevaluated the variance of thenumberof nodalintersections
under the assumption that the Fourier coefficient τ (4) is bounded away
m
from 1(i.e. τ boundedaway from thesingular measures 1(δ +δ )and
± m 4 ±1 ±i
itsπ/4-tilted version). Using[1]makes thatassumptionnolongernecessary.
c
2. An approximate Kac-Rice formula
2.1. The Kac-Rice premise. Let d 3, and Td be a smooth curve.
≥ C ⊆
Let F be the arithmetic random wave
1
F(x) = a e2πi µ,x
µ h i
√N
E
µ (E)
∈XE
(with the obvious generalization of all the previous notation to higher di-
mensions d 3). We wish to study the number (F) of nodal intersections
≥ Z
of F with by restricting F to as follows.
C C
Let γ : [0,L] R be a unit speed parameterization of . We restrict F
→ C
to by defining the random Gaussian process
C
(2.1) f(t)= F (γ(t))
on [0,L]. It is then obvious that the nodal intersections number (F) is
Z
equal to the number of the zeros of f. The Kac-Rice formula (see e.g. [8],
[1]) is a standard tool (meta-theorem) for evaluating the expected number
andhigher(factorial)momentsofzerosofa“generic”process: letX :I R
→
be a (a.s. C1-smooth, say) random Gaussian process on an interval I R,
⊆
and = the numberof zeros of X on I. For m 1 anddistinct points
I;X
Z Z ≥
t ,...,t I denote ϕ (u ,...u ) to be the (Gaussian) probability
1 m ∈ t1,t2,...,tm 1 m
densityfunctionoftherandomvector(X(t ),...X(t )) Rm. Then,under
1 m
∈
appropriate assumptions on X, the m-th factorial moment of is given by
Z
(2.2) E [m] = K (t ,...,t )dt ...dt ,
m 1 m 1 m
Z
h i ZIm
where
( 1) ( m+1) 1 m
[m] := Z Z − ··· Z − ≤ ≤ Z ,
Z (0 otherwise.
6 Z.RUDNICK,I.WIGMANANDN.YESHA
and K , given by
m
K (t ,...t )= ϕ (0,...,0)
m 1 m t1,...,tm ×
(2.3)
E X (t )...X (t ) X(t ) = 0,...,X(t ) = 0 ,
′ 1 ′ m 1 m
×
is the m-th zero-intensityh(cid:12)of . Note that fo(cid:12)r(cid:12)(cid:12)the Gaussian case i
(cid:12) (cid:12)
Z (cid:12)
1
(2.4) ϕ (0,...,0) =
t1,...,tm (2π)m/2√detA
with A the covariance matrix of the values (X(t ),...,X(t )), provided
1 m
that detA= 0, or, equivalently, that the distribution of (X(t ),...,X(t ))
1 m
6
is non-degenerate.
Thevalidity of themeta-theorem (2.2)was established undera numberof
various scenarios. Originally the result(2.2)was proven to hold [8] provided
that for all distinct points t ,...,t I the distribution of the Gaussian
1 m
∈
vector (X(t ),...,X(t ),X (t ),...X (t )) R2m is non-degenerate. This
1 m ′ 1 ′ m
∈
non-degeneracy condition was relaxed2 [1], as in the following theorem:
Theorem 2.1 ( [1], Theorem 6.3). Let X : I R be a Gaussian process
→
having C1 paths and m 1. Assume that for every m pairwise distinct
≥
points t ,...,t I, the joint distribution of (X(t ),...X(t )) Rm is
1 m 1 m
∈ ∈
non-degenerate. Then (2.2) holds.
The cases m = 1,2 are of our particular interest (see the following sec-
tions). Themain problemis that for m = 2even theweaker non-degeneracy
hypothesis in Theorem 2.1 may not be satisfied for our process f as in (2.1);
to resolve this issue we will decompose the interval I = [0,L] into small
subintervals, and apply Kac-Rice for each pair of the subintervals to de-
velop “approximate Kac-Rice” formula following an idea from [13] in the
two-dimensional case (see 2.4).
§
The covariance function of the centered Gaussian random field F reads
1
r (x,y) := E[F(x)F(y)] = cos(2π µ,y x )
F
N h − i
E
µ
X∈E
for x,y Td. As F is stationary (r depending on y x only), we may
F
∈ −
think of r as a function of one variable on Td. The covariance function of
F
f is
r(t ,t )= r (t ,t ) := E[f(t )f(t )] = r (γ(t ) γ(t )).
1 2 f 1 2 1 2 F 1 2
−
Therefore, f is a centered unit variance Gaussian process (non-stationary);
r(t ,t ) = 1 if and only if the joint distribution of f(t ),f(t ) is non-
1 2 1 2
6 ±
degenerate, so the probability density ϕ of the Gaussian random vector
t1,t2
2This fortunate fact simplifies our treatment of the approximate Kac-Rice formula
below (though it is possible to work [13] with the more restrictive version to obtain the
same results).
NODAL INTERSECTIONS FOR RANDOM WAVES ON THE 3-DIM TORUS 7
(f(t ),f(t )) exists. Denote
1 2
∂r ∂r ∂2r
r := ,r := ,r := ,
1 2 12
∂t ∂t ∂t ∂t
1 2 1 2
and let
(2.5) (E) := r2+(r /√E)2+(r /√E)2+(r /E)2 dt dt
2 1 2 12 1 2
R
[0,ZL]2 (cid:16) (cid:17)
be the sum of second moments of r and its few normalized derivatives along
; we will control the various quantities via (see Proposition 2.2 below).
2
C R
Later we will show that (E) is decaying with E ( 5).
2
R §
Proposition 2.2 (Approximate Kac-Rice formula). We have
Var Z = O( (E))
2
√E R
(cid:18) (cid:19)
with is given by (2.5).
2
R
Therestof thissection isdedicated to theproofof Proposition2.2,finally
given in 2.4, following some preparations.
§
2.2. Expectation. Since f is a centered unit variance Gaussian process
with C1 paths (in, particular, the non-degeneracy condition of Theorem 2.1
isautomatically satisfied),wemayusetheKac-Riceformula(2.2); form = 1
it reads
(2.6) E[ ]= K (t)dt
1
Z
Z
I
with
1
K (t)= E f (t) f(t)= 0 ,
1 ′
√2π ·
the “zero density” of f (here (2.4) reah(cid:12)ds ϕ ((cid:12)0(cid:12)(cid:12)) = 1 )i. Let Γ be the covari-
(cid:12) t(cid:12)(cid:12) √2π
ance matrix of (f(t),f (t)):
′
r(t,t) r (t,t)
Γ(t)= 1 .
r (t,t) r (t,t)
2 12
(cid:18) (cid:19)
Lemma 2.3. For a smooth curve of length L, the expectation of nodal in-
tersections number is given by
2
E[ ]= L √E
Z √d ·
Proof. Since f is unit variance, it is immediate that for every t [0,L] we
∈
have r (t,t) = r (t,t) = 0, so
1 2
1
Γ(t)= ,
α
(cid:18) (cid:19)
where
α:= r (t,t) = γ˙(t)tH (0,0)γ˙(t)
12 − rF
8 Z.RUDNICK,I.WIGMANANDN.YESHA
and H is the Hessian of r (see e.g. [13]). Since H (0,0) is a scalar
rF F rF
matrix [12]
4
H (0,0) = − π2E I
rF d · d
it follows that
4
α = π2E.
d
The distribution of f (t) conditional on f(t) = 0 is centered Gaussian with
′
variance α. Recall that for X N(0,σ2) we have E(X ) = σ 2/π, so
∼ | |
1 2 p
(2.7) K (t) = √α= √E
1
π √d
independentof x, and the statement of the lemma follows upon substituting
(2.7) into (2.6). (cid:3)
2.3. Variance. For m = 2 the Kac-Rice formula (2.2) reads
(2.8) E[ 2 ]= K (t ,t )dt dt
2 1 2 1 2
Z −Z
Z
I I
×
with K , the “2-point correlation function”, defined for r(t ,t ) = 1 as
2 1 2
6 ±
(see (2.3) and (2.4))
1
K (t ,t ) = E f (t ) f (t ) f(t ) = f(t )= 0 ,
2 1 2 ′ 1 ′ 2 1 2
2π√1 r2 · ·
− h(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) i
holding (Theorem 2.1) provided t(cid:12)hat fo(cid:12)r a(cid:12)ll t =(cid:12)(cid:12)t we have r(t ,t ) = 1
1 (cid:12)2 1 2
6 6 ±
(equivalently, the distribution of (f(t ),f(t )) is non-degenerate). Equiva-
1 2
lently, the zero number variance is given by (cf. (2.6))
(2.9) Var( ) = (K (t ,t ) K (t )K (t )) dt dt +E[ ].
2 1 2 1 1 1 2 1 2
Z − Z
Z
I I
×
Asitwas mentionedabove, inourcasetheassumptionthatr(t ,t )= 1
1 2
6 ±
for all t = t may not be satisfied, and, as explained in [13], it is easy to
1 2
6
construct an example of a curve where the Kac-Rice formula for the second
factorialmoment(2.8)doesnothold. Toresolve thissituation wewilldivide
the interval I = [0,L] into small subintervals, and note that the proof of [1]
Theorem 6.3 yields that if J ,J I are two disjoint subintervals, then
1 2
⊆
(recall that we denoted to be the number of zeros of f on a subinterval
J
Z
J I)
⊆
(2.10) E[ ] = K (t ,t )dt dt ,
ZJ1 ·ZJ2 2 1 2 1 2
Z
J1 J2
×
provided that for all t J , t J ,
1 1 2 2
∈ ∈
r(t ,t ) = 1.
1 2
6 ±
NODAL INTERSECTIONS FOR RANDOM WAVES ON THE 3-DIM TORUS 9
The 2-point correlation function was evaluated [13] explicitly to be
1
(2.11) K (t ,t )= µ ( 1 ρ2+ρarcsinρ),
2 1 2
π2(1 r2)3/2 · · −
−
p
where
(2.12) µ = µ (t ,t ) = α(1 r2) r2 α(1 r2) r2,
E 1 2 − − 1 · − − 2
q q
and
r (1 r2)+rr r
12 1 2
(2.13) ρ= ρ (t ,t )= −
E 1 2
α(1 r2) r2 α(1 r2) r2
− − 1 · − − 2
(it follows from the derivatiopn of (2.11) that ρp 1).
| | ≤
2.4. Proof of Proposition 2.2. Before giving the proof of Proposition 2.2
wewillhavetodosomepreparatorywork. Toovercometheabove-mentioned
obstacle we let c be a sufficiently small constant to be chosen below, and
0
decompose the interval [0,L] into small intervals of length roughly c 1
0 · √E
so that we can apply Kac-Rice on the corresponding diagonal cubes. To be
more concrete, let k = L √E + 1 and δ = L, and divide the interval
⌊ · c0 ⌋ 0 k
[0,L] into the subintervals I = [(i 1)δ ,iδ ] where i = 1,...,k. Note that
i 0 0
−
δ 1 . With denoting the number of zeros of f on I (i= 1,...,k) we
0 ≍ √E Zi i
have
(2.14) Var( ) = Cov( , ).
i j
Z Z Z
i,j
X
Our first goal is to give an upper bound for the individual summands in
(2.14);tothisendweneedthefollowinglemmas,whoseproofsarepostponed
till 2.5:
§
Lemma 2.4. There exists a constant c > 0 sufficiently small, such that
0
for all t = t [0,L] with t t < c /√E we have
1 2 2 1 0
6 ∈ | − |
r(t ,t ) = 1.
1 2
6 ±
Lemma 2.5 (Uniformboundonthe2-pointcorrelation functionaroundthe
diagonal). For all 0< t t < c /√E we have
2 1 0
| − |
K (t ,t ) = O(E).
2 1 2
Corollary 2.6. We have
Cov( , ) = O(1),
i j
Z Z
uniformly for all i,j and E (the implied constant is universal).
Proof of Corollary 2.6 assuming lemmas 2.4-2.5. ByLemma2.4,r(t ,t ) =
1 2
6
1 for all t = t in every diagonal cube I2. Hence (Theorem 2.1 applied on
± 1 6 2 i
10 Z.RUDNICK,I.WIGMANANDN.YESHA
the interval I corresponding to a diagonal cube I2) we can apply Kac-Rice
i i
(2.9) to compute the variance of :
i
Z
Var( ) = (K (t ,t ) K (t )K (t )) dt dt +E[ ].
i 2 1 2 1 1 1 2 1 2 i
Z − Z
Z
I2
i
By Kac-Rice we have
2
E[ ] = K (t)dt = δ √E;
i 1 0
Z · √d
Z
Ii
using Lemma 2.5 we conclude that
Var( ) Eδ2 +√Eδ 1.
Zi ≪ 0 0 ≪
This proves the statement of the corollary for i = j; the result for arbitrary
i,j follows from the above and the Cauchy-Schwartz inequality
Cov( , ) Var( ) Var( ).
i j i j
Z Z ≤ Z · Z
q (cid:3)
Definition 2.7. (Singular and nonsingular cubes.)
(i) Let
S = I I = [iδ ,(i+1)δ ] [jδ ,(j +1)δ ]
ij i j 0 0 0 0
× ×
be a cube in [0,L]2. We say that S is a singular if it contains a
ij
point (t ,t ) S satisfying
1 2 ij
∈
r(t ,t ) > 1/2.
1 2
| |
(ii) The union of all the singular cubes is the singular set
B = B = S .
E ij
Sij s[ingular
Note that since r/√E is a Lipschitz function with a universal constant
(independent of E), if S is a singular cube, then
ij
r(t ,t ) > 1/4
1 2
| |
everywhere on S , provided that c is chosen sufficiently small. Using the
ij 0
above it is easy to obtain the following bound on the number of singular
cubes:
Lemma 2.8. The number of singular cubes is bounded above by
E r2(t ,t )dt dt .
1 2 1 2
·
Z
[0,L]2