Small scale distribution of zeros and mass of modular forms

0.33 MB·English

by  Stephen Lester

SMALL SCALE DISTRIBUTION OF ZEROS AND MASS OF
MODULAR FORMS
STEPHEN LESTER, KAISA MATOMA¨KI, AND MAKSYM RADZIWIL L
5
1
0 Abstract. We study the behavior of zeros and mass of holomorphic Hecke cusp
2 forms on SL2(Z) H at small scales. In particular, we examine the distribution of
\
n the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k .
u We show that the zeros equidistribute within such balls as k as long a→s t∞he
J → ∞
radii shrink at a rate at most a small power of 1/logk. This relies on a new,
6
effective, proof of Rudnick’s theorem on equidistribution of the zeros and on an
1
effective version of Quantum Unique Ergodicity for holomorphic forms, which we

] obtain in this paper.
T WealsoexaminethedistributionofthezerosnearthecuspofSL2(Z) H. Ghosh
N \
andSarnakconjecturedthatalmostallthe zeroshere lieontwoverticalgeodesics.
. Weshowthatforalmostallformsapositiveproportionofzeroshighinthecuspdo
h
t lieonthesegeodesics. Forallforms,weassumetheGeneralizedLindelo¨fHypothesis
a
and establish a lower bound on the number of zeros that lie on these geodesics,
m
which is significantly stronger than the previous unconditional results.
[

2
v
2
9 1. Introduction
2
1 Let f be a modular form of weight k for SL (Z). A classical result in the theory of
2
0 modular forms states that the number of properly weighted zeros of f in SL (Z) H
1. equals k/12. Inside the fundamental domain = z H : 1/2 R2e(z)\<
0 F { ∈ − ≤
1/2, z 1 the distribution of the zeros of different modular forms of weight k can
5
| | ≥ }
1 vary drastically. For instance, F.K.C. Rankin and H.P.F. Swinnerton-Dyer [20] have
:
v proved that all the zeros of the holomorphic Eisenstein series
i
X 1 1
E (z) =
ar k 2 (cz +d)k
(c,d)=1
X
that lie inside lie on the arc z = 1 . Moreover, the zeros of E (z) are uniformly
k
F {| | }
distributed on this arc as k . In contrast, consider powers of the modular
→ ∞
ThesecondauthorwassupportedbytheAcademyofFinlandgrantsno. 137883and138522. The
research leading to these results has received funding from the European Research Council under
the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement
o
n 320755.
1
2 S.LESTER, K.MATOMA¨KI,AND M. RADZIWIL L
k
discriminant, that is, ∆(z)12 with 12 k. This function is a weight k cusp form and
|
has one distinct zero at with multiplicity k/12.
∞
The weight k Hecke cusp forms constitute a natural basis for the space of weight
k modular forms and the distribution of their zeros differs from the previous two
examples. Using methods from potential theory, Rudnick [21] showed that the ze-
ros of Hecke cusp forms equidistribute in the fundamental domain with respect
F
to hyperbolic measure in the limit as the weight tends to infinity. Rudnick’s result
originally relied on the then unproven Quantum Unique Ergodicity (QUE) conjec-
ture for holomorphic Hecke cusp forms. However this is now a theorem proved by
Holowinsky and Soundararajan [8] and so Rudnick’s result on the equidistribution
of zeros holds unconditionally.
It is natural to study what happens beyond equidistribution, and to investigate
the distribution of zeros and mass of Hecke cusp forms at smaller scales. That is, to
examine the behavior of the zeros and mass within sets whose hyperbolic area tends
to zero at a quantitative rate as the weight k . For the zeros, we consider the
→ ∞
following two different aspects of this problem:
1) ThedistributionofzerosofHeckecuspformswithinhyperbolicballsB(z ,r )
0 k
⊂
with r 0 sufficiently slowly as k .
k
F → → ∞
2) The distribution of the zeros of Hecke cusp forms in the domain
= z : Im(z) > Y Y klogk.
Y
F { ∈ F } ≥
The second problem also examines the zeros of f at a smpall scale since the hyperbolic
area of equals 1/Y andtends to zero as theweight tends to infinity. This problem
Y
F
was originally studied by Ghosh and Sarnak [3] who proved that many of the zeros of
f that lie inside lie oneach of thevertical geodesics Re(z) = 1/2 andRe(z) = 0.
Y
F −
Additionally, buildingonthetechniquesdevelopedbyHolowinskyandSoundarara-
jan we prove an effective form of QUE. Our result also applies to the small scale
setting and we show that the L2-mass of a weight k Hecke cusp equidistributes in-
side a rectangle whose hyperbolic area shrinks sufficiently slowly as k . This
→ ∞
complements recent work of Young [28] who studied QUE at even smaller scales un-
der the assumption of the Generalized Lindelo¨f Hypothesis. Notably, Young’s work
also applies to Hecke-Maass forms whereas the analog of our result for Hecke-Maass
forms is open.
1.1. Zeros of Hecke cusp forms in shrinking hyperbolic balls and effective
QUE. Two immediate difficulties appear when attempting to understand the dis-
tribution of zeros of Hecke cusp forms in shrinking hyperbolic balls: First of all, it is
not clear if it is possible to adapt Rudnick’s argument since it relies on a compact-
ness argument, which is not effective and does not apply to the small scale setting.
Secondly, the current results on QUE do not establish a rate of convergence. We
ZEROS OF MODULAR FORMS AND QUE 3
remedy the first difficulty by finding a new proof of Rudnick’s theorem, which is
effective. We address the second difficulty by revisiting the work of Holowinsky and
Soundararajan and extracting a rate of convergence from their result. This leads to
the following theorem.
Theorem 1.1. Let f be a sequence of Hecke cusp forms of weight k. Also, let
k
B(z ,r) be the hyperbolic ball centered at z and of radius r, with z fixed and
0 0 0
⊂ F
r (logk)−δ/2+ε where δ = 1 (31/2 4√15) = 0.001152.... Then as k ,we
≥ 7 · − → ∞
have
# ̺ B(z ,r) : f (̺ ) = 0 3 dxdy
{ f ∈ 0 k f } = +O r(logk)−δ/2+ε .
# ̺ : f (̺ ) = 0 π y2
{ f ∈ F k f } ZZB(z0,r) (cid:16) (cid:17)
This result is far from optimal since we expect equidistribution for the zeros of
Hecke cusp forms nearly all the way down to the Planck scale. That is, the zeros
of Hecke modular forms should equidistribute with respect to hyperbolic measure
within hyperbolic balls with area as small as k−1+ε. Assuming the Generalized
Lindelo¨f Hypothesis we can show this happens within hyperbolic balls with area as
small as k−1/4+ε.
Theorem 1.2. Assume the Generalized Lindel¨of Hypothesis. Let f be a sequence of
k
Hecke cusp forms of weight k. Also, let B(z ,r) be the hyperbolic ball centered
0
⊂ F
at z and of radius r, with z fixed and r k−1/8+ε. Then as k we have
0 0
≥ → ∞
# ̺ B(z ,r) : f (̺ ) = 0 3 dxdy
{ f ∈ 0 k f } = +O rk−1/8+ε .
# ̺ : f (̺ ) = 0 π y2
{ f ∈ F k f } ZZB(z0,r) (cid:16) (cid:17)
While QUE establishes that the mass of yk f(z) 2 equidistributes as the weight k
| |
of f grows, our proof of Theorem 2.1 shows that the equidistribution of the zeros
follows from the much weaker condition: For any fixed ε > 0 and for any fixed
domain , we have
R
dxdy
yk f(z) 2 e−εk.
·| | · y2 ≫
ZZR
We were not able to make use of this weaker condition, but remain hopeful that it
will be useful in later works (see Theorem 2.1 for precise results).
To understand the mass of f in shrinking sets we obtain the following effective
version of Quantum Unique Ergodicity in the holomorphic case.
Theorem 1.3 (Effective QUE). Let f be a Hecke cusp form of weight k. Then,
dxdy 3 dxdy
sup yk f(z) 2 (logk)−δ+ε
R⊂F(cid:12)(cid:12)ZZR | | y2 − π ZZR y2 (cid:12)(cid:12) ≪ε
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 S.LESTER, K.MATOMA¨KI,AND M. RADZIWIL L
with δ = 1 (31/2 4√15) = 0.001152... and where the supremum is taken over all
7 · −
the rectangles lying inside the fundamental domain that have sides parallel to
R F
the coordinate axes.
For general domains we cannot extract from the argument of Holowinsky and
R
Soundararajan a saving exceeding a small power of logk. However, assuming the
Generalized Lindelo¨f Hypothesis, Watson [26] and Young [28] have established a
power saving bound, which is an important ingredient in the proof of Theorem 1.2.
On the unconditional front, it was proven by Luo and Sarnak [14, 15] that one can
obtaincomparableresults onaverage, obtainingapower saving boundformost forms
f. Combining this input with our new proof of Rudnick’s theorem gives the following
variant of Theorem 1.1.
Theorem 1.4. Let be a Hecke basis for the set of weight k cusp forms. Let δ > 0.
k
H
Then, for all but at most k20/21+4δ forms f , we have for r k−δ/2
k
≪ ∈ H ≥
# ̺ B(z ,r) : f (̺ ) = 0 3 dxdy
{ f ∈ 0 k f } = +O rk−δ/2logk .
# ̺ : f (̺ ) = 0 π y2
{ f ∈ F k f } ZZB(z0,r) (cid:16) (cid:17)
1.2. Zeros of Hecke cusp forms in shrinking Siegel domains. Wealso consider
the distribution of the zeros of Hecke cusp forms within the set = z :
Y
F { ∈ F
Im(z) > Y with Y > √klogk. The hyperbolic area of equals 1, and Ghosh and
} FY Y
Sarnak [3] proved for a weight k Hecke cusp form, f , that k # ̺ k.
k Y ≪ { f ∈ FY} ≪ Y
They also observed that equidistribution should not happen here and conjectured
that almost all the zeros of f in lie on the vertical geodesics Re(z) = 1/2 and
k Y
F −
Re(z) = 0 with one half lying on each line.
In support of their conjecture Ghosh and Sarnak showed that many of the zeros
of f in lie on segments of the vertical lines Re(z) = 0 and Re(z) = 1/2. They
k Y
F −
proved that
(1.1) # ̺f Y : Re(̺f) = 0 or Re(̺f) = 1/2 (k/Y)21−410−ǫ.
{ ∈ F − } ≫
The term 1/40 in their result was subsequently removed in [16] by the second named
author.
In support of Ghosh and Sarnak’s conjecture, we establish the following result.
Theorem 1.5. Let ε > 0 be fixed. There exists a subset , containing more
k k
S ⊂ H
than (1 ε) elements, and such that every f we have
k k
− |H | ∈ S
# ̺ : Re(̺ ) = 0 c(ε) # ̺
f Y f f Y
{ ∈ F } ≥ · { ∈ F }
and
# ̺ : Re(̺ ) = 1/2 c(ε) # ̺
f Y f f Y
{ ∈ F − } ≥ · { ∈ F }
provided that δ(ε)k > Y > √klogk and k . The constants δ(ǫ) and c(ε) depend
→ ∞
only on ε.
ZEROS OF MODULAR FORMS AND QUE 5
The proof of Theorem 1.5 relies on a very recent result on multiplicative functions
by the second and third author [17]. For individual forms f we cannot do as well,
even on the assumption of the Lindelo¨f or Riemann Hypothesis. The reason is the
following: In order to produce sign changes of f we look at sign changes of the co-
efficients λ (n). In order to obtain a positive proportion of the zeros on the line we
f
need a positive proportion of sign changes between the coefficients of λ (n), in ap-
f
propriate ranges of n. However we cannot have a positive proportion of sign changes
if for example, for all primes p (logk)2−ε, we have λ (p) = 0. Unfortunately even
f
≤
on the Riemann Hypothesis we cannot currently rule out this scenario.
Nonetheless on the Lindelo¨f Hypothesis we can still obtain the following result,
which is significantly stronger than the previous unconditional result.
Theorem 1.6. Assume the Generalized Lindel¨of Hypothesis. Then for any ε > 0
(1.2) # ̺ : Re(̺ ) = 0 (k/Y)1−ε
f Y f
{ ∈ F } ≫
and
(1.3) # ̺ : Re(̺ ) = 1/2 (k/Y)1−ε,
f Y f
{ ∈ F − } ≫
provided that √klogk < Y < k1−δ for some δ > 0.
The paper is organized as follows: In Section 2 we investigate the results related to
equidistribution in shrinking sets. In Section 3 we prove the results on zeros high in
the cusp. Finally in Section 4 we establish the effective version of Quantum Unique
Ergodicity.
2. Zeros of cusp forms in shrinking geodesic balls
Let φ be a smooth function that is compactly supported within . Also, let D (z)
r
F
be the disk of radius r centered at z. Given a cusp form f, and a compact subset
, define,
R ⊂ F
dxdy
µ ( ) := yk f(z) 2 .
f R | | y2
ZZR
Here the form f is assumed to be normalized so that µ ( ) = 1. Also, let ∆ =
f
F
y2 ∂2 + ∂2 denote the hyperbolic Laplacian. The main component of the proofs
− ∂x2 ∂y2
of T(cid:16)heorems 1(cid:17).1, 1.2 and 1.4 is the following:
Theorem 2.1. Let f be a Hecke cusp form, and, z : Im(z) B where
R ⊂ { ∈ F ≤ }
B > 1. Also, let h(k) > logk and φ be a smooth compactly supported function in
k R
such that ∆φ h(k)−A for some A 0. Suppose for every z and k K(B)
0
≪ ≥ ∈ R ≥
there exists a point z = x +iy D (z ) satisfying
1 1 1 h(k) 0
∈
(2.1) yk f(z ) 2 e−kh(k).
1| 1 | ≫
6 S.LESTER, K.MATOMA¨KI,AND M. RADZIWIL L
Then,
k 3 dxdy
φ(̺ ) = φ(z) +O (k h(k)2)
f 12 · π y2 B ·
(2.2) X̺f ZZF
dxdy
+O k h(k)log1/h(k) ∆φ(z) .
A,B · | | y2
(cid:16) ZZF (cid:17)
By the QUE theorem of Holowinsky and Soundararajan (2.1) holds for fixed, but
arbitrarilysmall h(k). This reproduces themainresult ofRudnick [21]. Additionally,
Theorem 1.3 implies that (2.1) holds for h(k) (logk)−δ+ε with 1 (31/2 4√15) =
≫ 7· −
0.001152.... Assuming the Generalized Lindelo¨f Hypothesis it follows from an ar-
gument of Young [28] that (2.1) holds for h(k) k−1/4+ε. 1 Adapting some ideas
≥
from recent unpublished work of Borichev and Sodin to this setting it should be
possible to give a better estimate for the second error term in (2.2). In particular, a
consequence of this would improve the range of r in Theorem 1.1 to r (logk)−δ,
≥
whereas we require r (logk)−δ/2.
≥
Proof of Theorems 1.1 and 1.2. Let φ be a smooth function such that φ has com-
1 1
pact support within B(z ,r), φ (z) = 1 for z B(z ,r M−1), and ∆φ M2,
0 1 0 1
∈ − ≪
where M tends to infinity with k and will be chosen later. Also suppose that
r 2/M. Similarly, let φ be a smooth function such that φ has compact sup-
2 2
≥
port within B(z ,r +M−1,), φ (z) = 1 for z B(z ,r), and ∆φ M2. We have
0 2 0 2
∈ ≪
that
k 3 dxdy
12 · π |φ1(z)−φ2(z)| y2 ≪k ·AreaH(B(z0,r +M−1)\B(z0,r−M−1))
ZZF
k r M−1.
≪ · ·
Also ∆φ (z) dxdy rM. Next, observe that
F | j | y2 ≪
RR φ (̺ ) # ̺ B(z ,r) φ (̺ ).
1 f f 0 2 f
≤ { ∈ } ≤
X̺f X̺f
Thus, Theorem 2.1 implies
# ̺ B(z ,r) = k AreaH(B(z0,r))+O(rM k h(k)log1/h(k))+O(k r M−1).
f 0
{ ∈ } 12 AreaH( ) · · · ·
F
We take M = h(k)−1/2, h(k) = (logk)−δ+ε, in the unconditional case, and M =
h(k)−1/2,h(k) = k−1/4+ε in the conditional case. Using Theorem 1.3 then completes
the proof. The exponent δ is the same exponent as in Theorem 1.3. (cid:3)
1InProposition5.1of[28]Youngestablishesthe analogofthis forHecke-Maasscuspforms. The
proof for holomorphic case follows in much the same way.
ZEROS OF MODULAR FORMS AND QUE 7
For the proof of Theorem 1.4 we recall the work of Luo and Sarnak [14]. Define
the probability measure ν := (3/π)dxdy/y2 and denote by the space of Hecke
k
cusp forms for the full modular group SL (Z). Then, Luo andHSarnak (see Corollary
2
1.2 in [14]) showed that
1
(2.3) sup µ (B) ν(B) 2 k−1/21
f
# | − | ≪
k B
H fX∈Hk
where the supremum is taken over all geodesic balls B .
⊂ F
Proof of Theorem 1.4. For r k−1/2 let
1
≥
(r ) := f : z s.t. B(z ,r ) and z B(z ,r ), yk f(z) 2 k−2 .
k 1 k 0 0 1 0 1
E { ∈ H ∃ ⊂ F ∀ ∈ | | ≤ }
Notice that if f (r ) then we may apply Theorem 2.1 with h(k) r and
k k 1 1
∈ H \E ≪
argue as in the previous proof to get that for r √r
1
≥
k AreaH(B(z0,r))
# ̺ B(z ,r) = +O(rk √r log1/r ).
f 0 1 1
{ ∈ } 12 · AreaH( ) ·
F
It remains to bound the size of (r ). We apply (2.3) to see that
k 1
E
r4 # (r ) sup µ (B(z ,r )) ν(B(z ,r )) 2
1 · Ek 1 ≪ z0∈F | f 0 1 − 0 1 |
f∈E(r)
X
sup µ (B) ν(B) 2 k20/21,
f
≪ | − | ≪
fX∈Hk
where supremum in the second line is over all hyperbolic balls, B . The claim
⊂ F
follows taking r = k−δ. (cid:3)
1
2.1. Proof of Theorem 2.1. Let φ be a smooth function that is compactly sup-
ported on . Our starting point is the following formula of Rudnick (see Lemma 2.1
F
of [21], note that we assume φ is supported in )
F
k 3 dxdy 1 dxdy
(2.4) φ(̺ ) = φ(z) + log(yk/2 f(z) )∆φ(z) .
f 12 · π y2 2π | | y2
X̺f ZZF ZZF
To prove Theorem 2.1 we need to bound the second term in the above formula. The
difficulty here comes in estimating the contribution to the integral over the set where
f is exceptionally small.
We first require two auxiliary lemmas, the first of which is due to Cartan.
Lemma 2.2 (Theorem 9 of [12]). Given any number H > 0 and complex numbers
a ,a ,...,a , there is a system of circles in the complex plane, with the sum of the
1 2 n
8 S.LESTER, K.MATOMA¨KI,AND M. RADZIWIL L
radii equal to 2H, such that for each point z lying outside these circles one has the
inequality
H n
z a z a z a > .
1 2 n
| − |·| − |···| − | e
(cid:16) (cid:17)
For z = ̺ define
0 f
6
f(z)
M (z ) := max +3.
r 0
|z−z0|≤r(cid:12)f(z0)(cid:12)
The next lemma is from Titchmarsh [25] (s(cid:12)ee Lem(cid:12)ma α of section 3.9, especially
(cid:12) (cid:12)
formula (3.9.1)). (cid:12) (cid:12)
Lemma 2.3. Let g(z) be a holomorphic function on z z r, with g(z ) = 0.
0 0
| − | ≤ 6
Then there is an absolute constant A > 1 such that for z z r/4
0
| − | ≤
g(z) z ρ
log log − < AlogM (z ),
r 0
g(z ) − z ρ
(cid:12)(cid:12) (cid:12)(cid:12) 0 (cid:12)(cid:12) |ρ−Xz0|≤r/2 (cid:12)(cid:12) 0 − (cid:12)(cid:12)(cid:12)(cid:12)
where the summ(cid:12)ation(cid:12) runs(cid:12)over zeros ρ of(cid:12)g. (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12)
Let betheconvex hullofsuppφ. Letη,ε > 0. Wecover withN disksofradius
D D
ε centered at the points a ,...,a . The disks are chosen so that N Area( )/ε2.
1 N
≪ D
Define
= z : f(z)yk/2 < e−δk and = D (a ).
δ δ,j δ ε j
T { ∈ F | | } T T
Let n = # ̺ : ̺ D (a ) and set \
j f f 16ε j
{ ∈ }
ηε nj
= z D (a ) : z ̺ < .
η,j ε j f
S ∈ | − | e
n ̺f∈YD8ε(aj) (cid:16) (cid:17) o
By Cartan’s lemma the area of is 4πη2ε2.
η,j
S ≤
Lemma 2.4. Suppose that ε > logk/k and f(z )yk/2 e−ε·k. Then there exists a
0 0 ≫
constant C > 1 such that
M (z ) eCε·k.
16ε 0
≪
Proof. There is a point z = x +iy such that
max max max
k/2
f(z) f(z ) y k/2 y f(z )
max 0 max max
max = = .
z∈D16ε(z0)(cid:12)f(z0)(cid:12) (cid:12) f(z0) (cid:12) (cid:16)ymax(cid:17) ·(cid:12) y0k/2f(z0) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
By Proposition A.1 of(cid:12) Rudn(cid:12)ick(cid:12)[21] we h(cid:12)ave yk/2f(z (cid:12)) k1/2. ((cid:12)Note that Xia
(cid:12) (cid:12) (cid:12) (cid:12) max max(cid:12) (cid:12)
| | ≪
[27] has recently improved this bound to k−1/4+ǫ, but we do not need that here.)
≪
Also, yk/2f(z ) e−εk and
0 0 ≫
y k/2 y k/2
0 0 eCεk.
y ≤ y 16ε ≤
max 0
(cid:16) (cid:17) (cid:16) − (cid:17)
ZEROS OF MODULAR FORMS AND QUE 9
Combining these bounds we see that
M (z ) k1/2eε·k eCεk eC′εk.
16ε 0
≪ · ≪
(cid:3)
Lemma 2.5. Suppose ε > logk/k and that for all z there exists a point
0
∈ F
z = x + iy D (z ) such that yk f(z ) 2 e−εk. Then there is an absolute
1 1 1 ∈ ε 0 1| 1 | ≫
constant 1 > c > 0 such that for δ 1/c ε we have whenever η > exp( c δ/ε)
2 0 ≥ 0 · − 0
that
δ,j η,j
T ⊂ S
for each j = 1,...,N.
Proof. By assumption, for each j = 1,...,N there exists a point z D (a ) such
j ε j
∈
that f(z ) e−εky−k/2. If z then
| j | ≫ j ∈ Tδ,j
f(z) y k/2 y +2ε k/2
(2.5) j e−δk+εk e−δk+εk e−δk+3εk e−δk/4.
f(z ) ≪ y ≤ y ≤ ≤
(cid:12) j (cid:12)
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:12) (cid:12)
By Lemma(cid:12) 2.3 if(cid:12)z = ̺ there is a constant A > 1 such that for z z 1r
(cid:12) (cid:12) 0 6 f | − 0| ≤ 4
f(z) z ̺
0 f
log + log − < A logM (z ).
r 0
f(z ) z ̺ ·
(cid:12)(cid:12) (cid:12)(cid:12) 0 (cid:12)(cid:12) ̺f∈DXr/2(z0) (cid:12)(cid:12) − f (cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12)
Using this with(cid:12) z0 =(cid:12) zj al(cid:12)ong with (2.5) w(cid:12)e get tha(cid:12)t(cid:12) for z δ,j
∈ T
z ̺
j f
(2.6) AlogM (z ) < δk/5+ log − .
8ε j
− − z ̺
̺f∈XD4ε(zj) (cid:12)(cid:12) − f (cid:12)(cid:12)
(cid:12) (cid:12)
For z Dε(aj) η,j (cid:12) (cid:12)
∈ \S
(2.7)
z ̺ 4ε 4e 4e
log j − f log n log < A′logM (z )log ,
j 16ε j
z ̺ ≤ z ̺ ≤ η η
̺f∈XD4ε(zj) (cid:12)(cid:12) − f (cid:12)(cid:12) ̺f∈YD4ε(zj) | − f|
for some abso(cid:12)(cid:12)lute con(cid:12)(cid:12)stant A′ > 0 and the last inequality follows from Jensen’s
formula (we have also used the inequality z ̺ > z ̺
̺f∈D4ε(zj)| − f| ̺f∈D8ε(aj)| − f|
for z a < ε).
| − j| Q Q
For the sake of contradiction, suppose that is not contained in . Then
δ,j η,j
T S
combining (2.6) and (2.7) it follows that
δk
logM (z ) > .
16ε j 5(A+A′log4e/η)
However, by Lemma 2.4 logM (z ) εk, so that a contradiction is reached when
16ε j
≪
c is sufficiently small. (cid:3)
0
10 S.LESTER, K.MATOMA¨KI,AND M. RADZIWIL L
A simple consequence of the previous lemma gives us a bound on the size of our
exceptional set . This is one of the main ingredients in the proof of Theorem 2.1.
δ
T
Observe that under the hypotheses of the previous lemma
N N
(2.8) meas( ) meas( ) meas( ) N4π2η2ε2 η2.
δ δ,j η,j
T ∩D ≤ T ≤ S ≤ ≪
j=1 j=1
X X
We also require the following crude, yet sufficient bound on the second moment of
logyk/2 f(z) .
| |
Lemma 2.6. We have
(log(yk/2 f(z) ))2dxdy k2.
| | ≪
ZZD
Proof. Let c be as in Lemma 2.5. We take ε fixed but small, δ = 1/c ε and
0 0
·
η (exp( c δ/ε),1/2). For each j = 1,2,...,N (note that here N = O(1)) there
0
∈ −
exists c D (a ) such that c / , which by Lemma 2.5 implies that c / .
j ∈ ε j j ∈ Sη,j j ∈ T1/c0·ε,j
Thus, f(c ) e−1/c0·εk(Im(c ))−k/2 and c ̺ (εη/e)nj.
j ≫ j ̺f∈D8ε(cj)| j − f| ≥
Now apply Lemma 2.3 to see that for z c 2ε
Q| − j| ≤
f(z) z ̺
f
log = log − +O(logM (c )).
8ε j
f(c ) c ̺
(cid:12)(cid:12) j (cid:12)(cid:12) ̺f∈XD4ε(cj) (cid:12)(cid:12) j − f(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Apply Lemma 2.4 a(cid:12)nd ou(cid:12)r earlier observa(cid:12)tions to(cid:12)see that for z cj 2ε we have
| − | ≤
log f(z) = log z ̺ +O(k).
f
| | −
̺f∈XD4ε(cj) (cid:12) (cid:12)
(cid:12) (cid:12)
This implies that
(log f(z) )2dz n (log z ̺ )2dz +k2 k2.
j f
| | ≪ | − | ≪
Z|z−aj|≤ε ̺f∈XD4ε(cj)Z|z−cj|≤2ε
Summing over all the disks we see that
dxdy
(log(yk/2 f(z) ))2 k2 (logy)2dz + (log f(z) )2dz k2.
| | y2 ≪ | | ≪
ZZD ZD ZD
(cid:3)
We are now prepared to prove Theorem 2.1.
Proof of Theorem 2.1. By (2.4) it suffices to show that
1 dxdy dxdy
log(yk/2 f(z) )∆φ(z) k h(k)log1/h(k) ∆φ(z) +k h(k)2.
2π | | y2 ≪ · · | | y2 ·
(cid:12) ZZF (cid:12) ZZF
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)