Localized Quantitative Criteria for Equidistribution

0.19 MB·

by  Stefan Steinerberger

LOCALIZED QUANTITATIVE CRITERIA FOR EQUIDISTRIBUTION
STEFANSTEINERBERGER
7
1 Abstract. Let(xn)∞n=1 beasequenceonthetorusT(normalizedtolength1). Weshowthat
0 ifthereexistsasequenceofpositiverealnumbers(tn)∞n=1 convergingto0suchthat
n 2 Nl→im∞N12 mX,Nn=1√1tN exp(cid:18)−t1N(xm−xn)2(cid:19)=√π,
a
J sthizeeno(fxtnhe)∞ns=u1misisutnhiefnoremsslyendtiiastllryibduetteedr.mTinheidsiesxcelsupseicvieallylybyinltoecraesltginapgswahtesncatlNe ∼NN−−12.sTinhciesctahne
8 ∼
beusedtoshowequidistributionofsequenceswithPoissonianpaircorrelation,whichrecoversa
2
recentresultofAistleitner,Lachmann&PausingerandGrepstad&Larcher. Thegeneralform

oftheresultisprovenonarbitrarycompactmanifolds(M,g)wheretheroleoftheexponential
T] functionisplayedbytheheatkernelet∆: forallx1,...,xN M andallt>0
∈
h.N N12 mX,Nn=1[et∆δxm](xn)≥ vol(1M)
at andequalityisattainedasN →∞ifandonlyif(xn)∞n=1 equidistributes.
m
[

1 1. Introduction
v
1.1. Introduction. Let(x )∞ be a sequenceon[0,1]. Anaturallyassociatedobjectofinterest
3 n n=1
2 isthebehaviorofgapsonalocalscale. Ifthesequenceiscomprisedofindependentlyanduniformly
3 distributed random variables, then
8
1 s
0 lim # 1 m=n N : x x =2s almost surely.
m n
. N→∞N ≤ 6 ≤ | − |≤ N
1 Whenever a deterministinc sequence (x )∞ has the same poroperty, we say it has Poissonian pair
0 n n=1
correlation; this notion of pseudorandomness has been intensively investigated, see e.g. [5, 8, 9,
7
1 10]. Recently, Aistleitner, Lachmann & Pausinger [1] and Grepstad & Larcher [4] independently
: established that sequences with Poissonianpair correlation are uniformly distributed on [0,1].
v
Xi Theorem (Aistleitner-Lachmann-Pausinger[1],Grepstad-Larcher[4]). Let(xn)∞n=1 be asequence
on [0,1] and assume that for all s>0
r
a 1 s
lim # 1 m=n N : x x =2s a.s.,
m n
N→∞N ≤ 6 ≤ | − |≤ N
n o
then the sequence is uniformly distributed.
An intuitive explanation is that any type of clustering produces many pairs (x ,x ) which are
m n
close to each other – the two available proofs are very different; the proof in [4] also implies a
quantitative estimate on discrepancy. The purpose of this paper is to embed this result into an
entire family of criteria that imply uniform distribution – this family of criteria strenghten the
result cited above and are applicable on general compact manifolds.
1.2. Qualitative Results on the Torus. We startby formulating our resultin the special case
of the one-dimensional torus T (normalized to have length 1): let
∞
θ (x)=1+2 e−4π2n2tcos(2πnx) denote the Jacobi θ function.
t
−
n=1
X
2010 Mathematics Subject Classification. 11K06and42A16(primary),42A82and94A11(secondary).
Keywords and phrases. Uniformdistribution,paircorrelation,Jacobi thetafunction,crystallization.
1
2
We observe that θ (x) 0, θ (x)=θ (1 x),
t t t
≥ −
1 1/√t for x .√t
θ (x)dx=1 and θ (x) | |
t t
Z0 ∼(0 otherwise.
It looks roughly like a Gaussian centered at 0 with standard deviation t1/2 (the profile indeed
∼
converges to that of a Gaussian as t 0). Note that the property θ (x) =θ (1 x) implies that
t t
weneverhavetodistinguishbetween→pointsonthe unitinterval[0,1]andpoints−onthetorusTof
length 1. Our mainresult, when applied to the one-dimensionaltorus T, canbe stated as follows.
Corollary. Let (x )∞ be a sequence on T. If there exists a sequence (t )∞ of positive and
n n=1 n n=1
bounded real numbers such that
N
1
lim θ (x x )=1,
N→∞N2 tN n− m
m,n=1
X
then (x )∞ is uniformly distributed on T.
n n=1
Ifthesequence(t )∞ additionallyconvergesto0,thenwecansimplifythecriterionbyreplacing
n n=1
the Jacobi θ function with a suitable scaled Gaussian: indeed, if t 0, then
N
− →
N
1 1 1
lim exp (x x )2 =√π
N→∞N2 m,n=1√tN (cid:18)−tN m− n (cid:19)
X
immediately implies
N
1
lim θ (x x )=1.
N→∞N2 tN n− m
m,n=1
X
Figure 1. Highly localized Gaussians evaluated at neighboring points.
It is not difficult to see (and will be established as part of our argument) that for any 0 < s < t
and any x ,x ,...,x T
1 2 N
∈
N N
1 1
θ (x x ) θ (x x ) 1.
N2 s n− m ≥ N2 t n− m ≥
m,n=1 m,n=1
X X
This shows that the criterion becomes more restrictive if the sequence of scales (t )∞ is made
n n=1
smaller. Conversely, if that sequence is taken to be the constant sequence, t = 1 for all n N,
n
∈
the criterion becomes sharp and characterizes uniform distribution.
Corollary. A sequence (x )∞ is uniformly distributed on [0,1] or T if and only if
n n=1
N
1
lim θ (x x )=1.
N→∞N2 1 n− m
m,n=1
X
Thereisnothingspecialaboutt=1andtheresultholdsifθ isreplacedbyθ foranyfixedc>0.
1 c
Our proof gives a little bit more information and shows that we could replace θ by any function
1
φ C∞(T) satisfying
∈
φ(x)dx=1, φ(x)=φ(1 x) and φ(x)e2πikxdx>0 for all k N.
T − T ∈
Z Z
This result is related to the classical Bochner-Herglotz theorem and variants exist on other topo-
logical groups (see e.g. [6]). We emphasize that our Corollary, using the Jacobi θ function, is
−
3
a consequence of a result on general compact manifolds that does not assume any type of group
structure.
1.3. Application to pair correlation. The result cited above states that if for all s>0
1 s
lim # 1 m=n N : x x =2s,
m n
N→∞N ≤ 6 ≤ | − |≤ N
n o
then the sequence is uniformly distributed. A natural question is whether it is truly necessary to
require the limit relation to hold for all s>0. We can use our criterion from above to show that
it suffices to know it for all s N (slightly sharper results could be obtained but this is not the
∈
focusofthispaper). Thisshouldstillbefarfromoptimalandsharpercriteriacouldbeofinterest.
Corollary (Pair correlation). Let (x )∞ be a sequence on [0,1] and assume that for all s N
n n=1 ∈
1 s
lim # 1 m=n N : x x =2s,
m n
N→∞N ≤ 6 ≤ | − |≤ N
n o
then the sequence is uniformly distributed.
The criterion cannot be directly applied to use pair correlation: for that we would be required to
work on the scale t N−2 and it is not too difficult to see that this is where the criterion has
N
∼
to stop working because the diagonal terms are already too large
N N
1 1 θ (0) 1
θ (x x ) θ (x x )= tN &1.
N2 tN n− m ≥ N2 tN n− n N ∼ Nt1/2
m,n=1 n=1 N
X X
However,itisfairlyeasytoseethatitispossibletomakeaslightadaptiontothescale,essentially
t f(N)N−2 for some suitably chosen and very slowly growing unbounded sequence f(N)
N
∼
(growing at a speed depending on the speed of convergence of the pair correlation function).
Many variants are conceivable, we prove the following natural generalization.
Corollary (WeakPaircorrelation). Let (x )∞ bea sequenceon [0,1],let 0<α<1and assume
n n=1
that for all s>0
1 s
lim # 1 m=n N : x x =2s a.s.,
N→∞N2−α ≤ 6 ≤ | m− n|≤ Nα
n o
then the sequence is uniformly distributed.
We emphasize that this result is more widely applicable since the requirementof weak pair corre-
lationis less stringent. Consider,for example,asequence (x )∞ obtainedby takingx to be
n n=1 2n−1
i.i.d. uniformly chosen random variables on [0,1] and x = x . This sequence does not have
2n 2n−1
Poissonian pair correlation but satisfies the notion of weak pair correlation for all 0 < α < 1 a.s.
The argument naturally generalizes to other geometric spaces and can be roughly summarized as
saying that whenever
# 1 m,n N : x x s t for some t 0 and all s>0
m n N N
{ ≤ ≤ k − k≤ · } →
behaves as it would for Poissonian random variables, then the sequence is uniformly distributed
sincecriteriaofthistypecanbeusedtodeterminethevalidityofthelimitrelationinourcriterion.
1.4. A quantitative result. Ourmethodisflexible enoughtoallowforthe derivationofquanti-
tative results. We only discuss the simplest case,the method applies to fairly generaldiscrepancy
systems on compact manifolds. Let now x ,x ,...,x T. Discrepancy is defined as the maxi-
1 2 N
mum deviation of uniform and empirical distribution on∈the set of all intervals J T
⊂
# 1 i N :x J
i
D ( x ,...,x )= sup { ≤ ≤ ∈ } J .
N 1 N
{ } J⊂T(cid:12) N −| |(cid:12)
(cid:12) (cid:12)
ItiseasytoseethatDN tendsto0asN (cid:12)ifandonlyifthesequenceisu(cid:12)niformlydistributed.
We recall that for all x ,...,x T →∞(cid:12) (cid:12)
1 N
∈
N
1
θ (x x ) 1 and decreases monotonically in t.
N2 t n− m ≥
m,n=1
X
4
Corollary (Discrepancy bound). There exists a universal constant c > 0 such that for any
x ,x ,...,x T
1 2 N
∈
N
1
D2 c θ (x x ) 1
N ≤ N2 m,n=1 −cloDgN2DN n− m − !
X
We believe the result to be somewhat amusing but do not know whether it can be useful in a
more generalcontext. We note that the θ function operateson spatialscale D (logD )−1/2.
N N
− ∼
Similar results can be obtained on general manifolds using the same argument. We point out a
connection to crystallization problems: given N points on a manifold interacting via a nonlocal
energy,minimizing configurationoften arrangethemselves into periodic structures (we refer to [3]
for an introduction and to [2, 7] for results involving the Jacobi θ function).
−
1.5. The general result. Let (M,g) be a smooth compact manifold. We use et∆ to denote the
heat kernel, i.e. the semigroup that allows to solve the heat equation
(∂ ∆ )et∆u =0 on M [0, ]
t g 0
− × ∞
andwillapplyitmostlytoDiracδfunctionslocatedonthemanifold. Notethatclassicalshort-time
asymptotics show
d(x,y)2
et∆δ (y) t−d/2exp ,
x
∼ − 4t
(cid:18) (cid:19)
where d(x,y) is the geodesic d(cid:2)istance(cid:3): we are therefore, for t sufficiently small, essentially dealing
with Gaussians centered at x. We will prove the general inequality
N
1 1
et∆δ (x )
N2 xm n ≥ vol(M)
m,n=1
X (cid:0) (cid:1)
and show that asymptotic sharpness characterizes uniform distribution of (x )∞ .
n n=1
Figure 2. et∆δ (y) behaves like a Gaussian centered at y and scale √t.
x
∼
(cid:2) (cid:3)
Theorem. Let (M,g) be a smooth compact manifold and let (x )∞ be a sequence on M. If
n n=1
there exists a bounded sequence of times 0<t C such that
N
≤
N
1 1
lim etN∆δ (x )= ,
N→∞N2 xm n vol(M)
m,n=1
X (cid:0) (cid:1)
then the sequence is uniformly distributed on (M,g). Moreover, if t = c > 0 is constant, then
N
this limit relation holds if and only if (x )∞ is uniformly distributed.
n n=1
It is immediately clear that on special manifolds such as Td or Sd−1 (where explicit formulas for
the heat kernel are available), the result could be simplified and put into a similar form as the
corollariesaboveon[0,1]orT. Itisalsonotdifficulttoseethattheconditioninthe Theoremcan
never be satisfied if t decays faster than N−2/d since
N
N N
1 1 1
etN∆δ (x ) etN∆δ (x )& t−d/2,
N2 xm n ≥ N2 xm m N N
m,n=1 m=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
5
whichis 1fort N−2/d. We quicklysketchwhathappensifweapplythe resultonthe torus:
N
∼ ∼
the heat kernel has the explicit form
[et∆(δ )](y)=θ (x y).
x t
−
The comparison between the Jacobi θ function θ (x) for t small comes from the asymptotic
t
−
estimate
1 x2
θ (x) exp .
t
∼ √4πt −4t
(cid:18) (cid:19)
It is easy to see the incurrederroris small (indeed, many orderssmaller than wouldbe required).
2. Proof of the Main Theorem
2.1. Warmingup. Beforegoingintodetails,wegiveaverysimpleargumentforthemonotonicity
formula in the simplest case and prove that for any 0<s<t and any x ,x ,...,x T
1 2 N
∈
N N
1 1
θ (x x ) θ (x x ) 1.
N2 s n− m ≥ N2 t n− m ≥
m,n=1 m,n=1
X X
The proof is straightforward: first, we rewrite the expression as
N N N
1 1 1
θ (x x )= θ δ δ dx.
N2 s n− m T s∗ N xn! N xn!
m,n=1 Z n=1 n=1
X X X
The Plancherel identity yields
\ \
N N N N
1 1 1 1
θ δ δ dx= θ δ (ℓ) δ (ℓ)dx.
ZT s∗ N n=1 xn! N n=1 xn! ℓ∈Z s∗ N n=1 xn! N n=1 xn!
X X X X X
We note that
\ \
N N
1 1
θ δ (ℓ)=θ (ℓ) δ (ℓ)
s∗ N xn! s N xn!
n=1 n=1
X X
b
and note that we can expand
\
N
1
δ = a e2πiℓx
N xn ℓ
n=1 ℓ∈Z
X X
with a =a and a =1. We also note that
ℓ −ℓ 0
θ = e−4π2ℓ2te2πiℓx
s
ℓ∈Z
X
b
and thus
N N
1 1
θ δ δ dx= e−4π2ℓ2t a 2 =1+ e−4π2ℓ2t a 2,
ZT s∗ N n=1 xn! N n=1 xn! ℓ∈Z | ℓ| ℓ∈Z | ℓ|
X X X Xℓ6=0
which is monotonically decreasing in t. On general manifolds, we will repeat the argument with
the Fourierbasis of L2 being replacedby eigenfunctions of ∆ . The semigrouppropertiesof the
g
−
heat kernel serve as a substitute for the behavior of convolution under Fourier transform.
6
2.2. Structure of the Argument. Wequicklyoutlinetheoverallstructureoftheargumentand
will then divide the proof accordingly. The proof has five steps.
(1) We will start by showing that
N
1
et∆δ (x ) is monotonically decreasing in t.
N2 xm n
m,n=1
X (cid:0) (cid:1)
Since we are dealing with a bounded sequence of times 0<t C, monotonicity implies
N
≤
that it suffices to prove the main result only for t =C.
n
(2) If the sequence is not uniformly distributed, there exists a ball B and ε>0 such that for
infinitely many N there are (B +ε)N/vol(M) out of the first N elements contained in
| |
the ball B. We then consider, for a sufficiently small but fixed time δ = δ > 0, the
B,ε
function
N
1
eδ∆ δ
N xm
m=1
X
and prove that the average value of the function in a small neighborhood of B is bigger
than vol(M)−1+c for some fixed c >0 and infinitely many N N.
ε,δ,B ε,δ,B
∈
(3) The Cauchy-Schwarz inequality then implies
N
1 1
eδ∆ δ c∗ for infinitely many N N.
(cid:13) N xm − vol(M)(cid:13) ≥ ε,δ,B ∈
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
(4) Weuse(cid:13)(cid:13)thespectraltheorem,the(cid:13)(cid:13)eigenfunctions(φk)∞k=1 oftheLaplace-Beltramioperator
∆ andinequalitiesrelatedtothe compactnesseδ∆ :L1(M) Hs(M) toconclude that
g
−there exists a constant N N, depending only on B,ε,δ s→uch that for all N and all
0
∈
x ,...,x M
1 N
∈
2 2
N N
1 1 1 1 1
eδ∆ δ ,φ eδ∆ δ .
(cid:12)* N xm − vol(M) k+(cid:12) ≥ 2(cid:13) N xm − vol(M)(cid:13)
kX≤N0(cid:12) mX=1 (cid:12) (cid:13) mX=1 (cid:13)L2(M)
(cid:12) (cid:12) (cid:13) (cid:13)
Combi(cid:12)(cid:12)ning these last two steps implies(cid:12)(cid:12)the exi(cid:13)(cid:13)stence of infinitely many N(cid:13)(cid:13) N such that
∈
2
N
eδ∆ 1 δ 1 ,φ 1 c∗ 2 >0
(cid:12)* N xm − vol(M) k+(cid:12) ≥ 2 ε,δ,B
kX≤N0(cid:12)(cid:12) mX=1 (cid:12)(cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
and, finally, we ca(cid:12)n use this to show that (cid:12)
2
N
1 1
eC∆ δ ,φ c∗∗ >0 for infinitely many N N.
(cid:12)* N xm − vol(M) k+(cid:12) ≥ ε,δ,B,N0,C ∈
kX≤N0(cid:12) mX=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(5) We(cid:12) conclude by arguing that (cid:12)
2
N N N
1 1 1 1
e2C∆ δ , δ + eC∆ δ ,φ
* N xm N xm+≥ vol(M) (cid:12)* N xm k+(cid:12)
mX=1 mX=1 kX≤N0(cid:12) mX=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
from which the result then follows when using that (cid:12) (cid:12)
N N
1 1 1
eC∆ δ ,φ = eC∆ δ ,φ for all k 1
* N xm − vol(M) k+ * N xm k+ ≥
m=1 m=1
X X
because these eigenfunctions are orthogonalto φ (x)=vol(M)−1/2.
0
7
2.3. Proof of Theorem 2.
Proof. Let (M,g) be given and let us consider the L2 normalized Laplacian eigenfunctions
−
∆ φ =λ φ
g n n n
−
as a basis of L2(M). Observe that λ =0 and φ =vol(M)−1/2 is a constant function.
0 0
Step 1. We can rewrite the expression as
N N N
1 1 1
etN∆δ (x )= etN∆ δ , δ .
N2 xm n N xm N xm
* ! !+
m,n=1 m=1 m=1
X (cid:0) (cid:1) X X
This particularalgebraicstructure behaves wellunder the heat flow: for any f C∞(M), we can
∈
write
∞ ∞
f = f,φ φ and et∆f = e−λkt f,φ φ
k k k k
h i h i
k=0 k=0
X X
and thus
∞
et∆f,f = e−λkt f,φ 2.
k
h i
k=0
(cid:10) (cid:11) X
This quantity is obviously monotonically decreasing in t. Note that
2 2
1 1
lim et∆f,f = f,φ 2 = f, = fdg ,
t→∞ h 0i vol(M)1/2 vol(M)
(cid:28) (cid:29) (cid:18)ZM (cid:19)
(cid:10) (cid:11)
which immediately implies, using a density argument, that for all t>0
N N
1 1 1
et∆ δ , δ .
* N xm! N xm!+≥ vol(M)
m=1 m=1
X X
Step 2. Let us now assume that the sequence (x )∞ is not uniformly distributed but that
n n=1
nonetheless
N
1 1
lim etN∆δ (x )= .
N→∞N2 xm n vol(M)
m,n=1
X (cid:0) (cid:1)
The monotonicity of the expression under the heat flow implies that also
N
1 1
lim eC∆δ (x )= ,
N→∞N2 xm n vol(M)
m,n=1
X (cid:0) (cid:1)
whereC istheuniformupperboundonthesequenceoftimest . Notbeinguniformlydistributed
N
means there exists a geodesic ball B M and ε >0 such that
0
⊂
# 1 m N :x B B
m
{ ≤ ≤ ∈ } | | ε for infinitely many N.
0
N − vol(M) ≥
(cid:12) (cid:12)
(cid:12) (cid:12)
We shall assum(cid:12) e that (cid:12)
(cid:12) (cid:12)
# 1 m N :x B B
m
{ ≤ ≤ ∈ } | | ε for infinitely many N
0
N − vol(M) ≥
because the other case implies the same estimate for another ball B′ M B (possibly with a
⊆ \
different value of ε ). We may assume without loss of generality (by possibly making ε smaller)
0 0
that B 1/2. Let B denote the δ neighborhood of B and let δ >0 be chosen so small that
δ
| |≤ −
B ε
δ 0
| | 1+
B ≤ 100
| |
and let t >0 be chosen so small that
0
ε
inf et0∆δ (x)dx 1 0 .
z
z∈BZBδ ≥ − 100
(cid:2) (cid:3)
8
These two facts imply that for infinitely many N
N
1 ε 1
et0∆ δ (x)dx 1 0 (B +ε ) .
ZBδ" N mX=1 xm# ≥(cid:16) − 100(cid:17) | | 0 vol(M)
This means that the averagevalue satisfies
N
1 1 ε B +ε 1
et0∆ δ (x)dx 1 0 | | 0
|Bδ|ZBδ" N mX=1 xm# ≥(cid:16) − 100(cid:17) |Bδ| vol(M)
ε B ε 1
0 0
1 | | +
≥ − 100 B B vol(M)
(cid:16) (cid:17)(cid:18)| δ| | δ|(cid:19)
ε ε 1
0 0
1 1 +ε
0
≥ − 100 − 100 vol(M)
(cid:16) (cid:17)(cid:16) (cid:17)
98ε 99ε2 1 1
1+ 0 0 >
≥ 100 − 10000 vol(M) vol(M)
(cid:18) (cid:19)
Step 3. The Cauchy-Schwarzinequality implies, for general functions f that
1 1 1
B fdx = f dx
δ
| | B − vol(M) − vol(M)
(cid:12)| δ|ZBδ (cid:12) (cid:12)ZBδ(cid:18) (cid:19) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) f 1 2dx(cid:12)(cid:12) 1/2 B 1/2
δ
≤ ZBδ(cid:18) − vol(M)(cid:19) ! | |
andthereforethereexistsaconstantε >0(dependingonlyonε and B )suchthatforinfinitely
1 0 δ
many N N | |
∈
N
1 1
et0∆ δ (x) ε .
(cid:13)" N xm# − vol(M)(cid:13) ≥ 1
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
Step 4. We will now prove(cid:13)that for any t >0, any N N(cid:13)and any set of points x ,x ,...,x
(cid:13) 0 ∈ (cid:13) 0 1 N
N
1
et0∆ δ . 1
(cid:13)∇ N xm(cid:13) t0,(M,g)
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
for some implicit constant tha(cid:13)t is both independ(cid:13)ent of N and the actual set of points. It is easy
(cid:13) (cid:13)
to see that
N N
1 1
et0∆ δ = et0∆δ
(cid:13)∇ N xm(cid:13) (cid:13)N ∇ xm(cid:13)
(cid:13) mX=1 (cid:13)L∞(M) (cid:13) mX=1 (cid:13)L∞(M)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13)1 N (cid:13)
(cid:13) (cid:13) (cid:13) et0∆δ (cid:13)
≤ N ∇ xm L∞(M)
m=1
X (cid:13) (cid:13)
sup (cid:13)et0∆δ (cid:13) . 1,
≤ ∇ x L∞(M) (M,g),t0
x∈M
(cid:13) (cid:13)
which follows from the regularity of the Green’s fun(cid:13)ction. By(cid:13)the same token
N N
1 1
1= et0∆ δ vol(M)1/2 et0∆ δ
(cid:13) N xm(cid:13) ≤ (cid:13) N xm(cid:13)
(cid:13) mX=1 (cid:13)L1(M) (cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) 1 N(cid:13) (cid:13)
(cid:13) (cid:13) . (cid:13) et0∆δ (cid:13)
(M,g) (cid:13)N xm(cid:13)
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
sup (cid:13)et0∆δ .(cid:13) 1.
≤ (cid:13) x L2(M) (cid:13)(M,g),t0
x∈M
(cid:13) (cid:13)
(cid:13) (cid:13)
9
We can now use the spectral theorem to write
2 2
N ∞ N
1 1
1& et0∆ δ = λ et0∆ δ ,φ .
(M,g),t0 (cid:13)∇ N xm(cid:13) k(cid:12)* N xm k+(cid:12)
(cid:13) mX=1 (cid:13)L2(M) Xk=0 (cid:12) mX=1 (cid:12)
(cid:13) (cid:13) (cid:12) (cid:12)
However, at the same ti(cid:13)me, the eigenvalue(cid:13)s of the Laplace(cid:12)-Beltrami operator are(cid:12)monotonically
(cid:13) (cid:13) (cid:12) (cid:12)
increasing and unbounded
0<λ <λ .
1 2
≤···→∞
Weyl’s law would give the asymptotic growth but that is not necessary. Recall that we have that
for infinitely many N N
∈
N
1 1
et0∆ δ (x) ε .
(cid:13)" N xm# − vol(M)(cid:13) ≥ 1
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
We can now argue that fo(cid:13)r N 1 (cid:13)
(cid:13) 1 ≥ (cid:13)
2
N
1 1
ε2 et0∆ δ (x)
1 ≤(cid:13)" N xm# − vol(M)(cid:13)
(cid:13) mX=1 (cid:13)L2(M)
(cid:13) (cid:13)
(cid:13) N 2(cid:13) N 2
(cid:13) 1 (cid:13) 1
= et0∆ δ ,φ + et0∆ δ ,φ
(cid:12)* N xm k+(cid:12) (cid:12)* N xm k+(cid:12)
0<λXk≤N1(cid:12) mX=1 (cid:12) λkX>N1(cid:12) mX=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) N (cid:12)2 (cid:12) N (cid:12) 2
(cid:12) 1 (cid:12) 1 (cid:12) 1 (cid:12)
et0∆ δ ,φ + λ et0∆ δ ,φ
≤ (cid:12)* N xm k+(cid:12) N1 k(cid:12)* N xm k+(cid:12)
0<λXk≤N1(cid:12) mX=1 (cid:12) λkX>N1 (cid:12) mX=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) N (cid:12)2 (cid:12) N 2 (cid:12)
(cid:12) 1 (cid:12) 1 1(cid:12) (cid:12)
et0∆ δ ,φ + et0∆ δ .
≤ (cid:12)* N xm k+(cid:12) N1 (cid:13)∇ N xm(cid:13)
0<λXk≤N1(cid:12) mX=1 (cid:12) (cid:13) mX=1 (cid:13)L2(M)
(cid:12) (cid:12) (cid:13) (cid:13)
Step4implies thatthis(cid:12)finalgradienttermis u(cid:12)niformly(cid:13)boundedforallN(cid:13)andall x ,...,x T.
This means, that there(cid:12) exists N N and ε (cid:12)> 0 dep(cid:13)ending only on (M(cid:13),g),t ,ε 1such thNat∈, for
1 2 0 1
∈
infinitely many N,
2
N
1
et0∆ δ ,φ ε .
(cid:12)* N xm k+(cid:12) ≥ 2
0<λXk≤N1(cid:12) mX=1 (cid:12)
Step 5. Using representation in Fourie(cid:12)(cid:12)r series, we easily get t(cid:12)(cid:12)hat for all s>t0
(cid:12) (cid:12)
2
N
1
(cid:12)*es∆N δxm,φk+(cid:12) ≥e−N12(s−t0)ε2.
0<λXk≤N1(cid:12) mX=1 (cid:12)
We conclude by arguing that, for(cid:12)(cid:12)infinitely many N N(cid:12)(cid:12),
(cid:12) ∈ (cid:12)
N N N N
1 1 1 1
e2C∆ δ , δ = eC∆ δ ,eC∆ δ
N xm N xm N xm N xm
* + * +
m=1 m=1 m=1 m=1
X X X X
2
∞ N
1
= eC∆ δ ,φ
(cid:12)* N xm k+(cid:12)
kX=0(cid:12) mX=1 (cid:12)
(cid:12) (cid:12) 2
(cid:12) ∞ N(cid:12)
= 1(cid:12) + eC∆ 1 (cid:12) δ ,φ
vol(M) (cid:12)* N xm k+(cid:12)
kX=1(cid:12) mX=1 (cid:12)
(cid:12) (cid:12) 2
(cid:12) N (cid:12)
1 + (cid:12) eC∆ 1 δ ,φ (cid:12)
≥ vol(M) (cid:12)* N xm k+(cid:12)
kX≤N0(cid:12) mX=1 (cid:12)
(cid:12) (cid:12)
1 +e−N12((cid:12)(cid:12)C−t0)ε2. (cid:12)(cid:12)
≥ vol(M)
10
(cid:3)
3. Proof of the Discrepancy Bound
Lemma. Let t>0 and consider θ :T R given by.
t +
→
∞
θ (x)=1+2 e−4π2n2tcos(2πnx).
t
n=1
X
If ε>0 and
x
x 2 log(2/ε)√t, then θ (y)dy 1 ε.
t
≥ ≥ −
Z−x
p
Proof. A simple topological argument allows to compare the heat kernel on the torus with the
heat kernel on the real line: on the torus we have the possibility of looping around which we do
not have on the real line. Therefore, for all x R and all t>0
∈
∞
1+2 e−4π2n2tcos(2πnx) 1 e−x42t.
≥ √4πt
n=1
X
The result then follows from the Chernoff bound
∞ 1 e−y22dy e−x22.
√2π ≤
Zx
(cid:3)
Proof of the Discrepancy Bound. Let x ,...,x T be given and assume that
1 N
∈
D ( x ,...,x )=ε.
N 1 N
{ }
Then there exists an interval J T such that
⊂
# 1 m N :x J
m
{ ≤ ≤ ∈ } J =ε.
N −| |
(cid:12) (cid:12)
(cid:12) (cid:12)
We distinguish two cases: (cid:12) (cid:12)
(cid:12) (cid:12)
# 1 m N :x J # 1 m N :x J
m m
{ ≤ ≤ ∈ } = J +ε and { ≤ ≤ ∈ } = J ε.
N | | N | |−
We start with the first case, the second case is essentially identical. We set
1 ε2
t= .
100log(20/ε)
This choice guarantees, using the Lemma above, that
ε/4 ε
θ (y)dy 1 .
t
≥ − 10
Z−ε/4
We will now consider the slightly larger interval J∗ given as the ε/4 neighborhood of J. We see
that, for infinitely many N N, −
∈
N N
1 1 ε 1 1
e(t/2)∆ δ (x) 1 δ
J∗ (cid:13)" N xm# (cid:13) ≥ − 10 J∗ (cid:13)N xm(cid:13)
| |(cid:13) mX=1 (cid:13)L1(J∗) (cid:16) (cid:17)| |(cid:13) mX=1 (cid:13)L1(J)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) ε 1(cid:13) 1 N(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
1 δ
≥ − 10 J +ε/2(cid:13)N xm(cid:13)
(cid:16) (cid:17)| | (cid:13) mX=1 (cid:13)L1(J)
(cid:13) (cid:13)
ε J +ε (cid:13) (cid:13)
1 | | (cid:13) (cid:13)
≥ − 10 J +ε/2
(cid:16) (cid:17)| |
ε 1+ε
1
≥ − 10 1+ε/2
(cid:16) ε (cid:17)
1+ .
≥ 10