CORNERS IN DENSE SUBSETS OF Pd
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
A BSTRACT. Let Pd be the d-fold direct product of the set of primes. We prove that if A is a subset of Pd of
positive relative upper density then A contains infinitely many “corners”, that is sets of the form
{x, x + te1 , ..., x + ted }
where x ∈ Zd , t ∈ Z and {e1 , .., ed } are the standard basis vectors of Zd . The main tools are the hypergraph removal lemma, the linear forms conditions of Green-Tao and the transference principles of Gowers and
Reingold et al.
1. I NTRODUCTION
A remarkable result in additive number theory due to Green and Tao [8] proves the existence of arbitrary
long arithmetic progressions in the primes. It roughly states that if A is a subset of the primes of positive
relative upper density then A contains arbitrary constellations, that is non-trivial affine copies of any finite
set of integers. It might be viewed as a relative version of Szemerédi’s theorem [20] on the existence of long
arithmetic progressions in dense subsets of the integers. In higher dimensions, the multi-dimensional extension of Szemerédi’s theorem first proved by Furstenberg and Katznelson [4], which states that if A ⊆ Zd is
of positive upper density then A contains non-trivial affine copies of any finite set F ⊆ Zd . The proof in
[4] uses ergodic methods however a more recent combinatorial approach was developed by Gowers [5] and
also independently by Nagel, Rödl and Schacht [15].
It is natural to ask if both results have a common extension, that is if the Furstenberg-Katznelson theorem can be extended to subsets of Pd of positive relative upper density, that is when the base set of integers
are replaced by that of the primes. In fact, this question was raised by Tao [22], where the existence of
arbitrary constellations among the Gaussian primes was shown. A partial result, extending the original approach of [8], was obtained earlier by B. Cook and the first author [3], where it was proved that relative
dense subsets of Pd contain an affine copy of any finite set F ⊆ Zd which is in general position, meaning
that each coordinate hyperplane contains at most one point of F .
However when the set F is not in general position, it does not seem feasible to find a suitably pseudorandom measure supported essentially on the d-tuples of the primes, due to the self-correlations inherent
in the direct product structure. For example, if we want to count corners {(a, b), (a + d, b), (a, b + d)} in
A ⊆ P2 then if (a + d, b), (a, b + d) ∈ P2 then the remaining vertex (a, b) must also be in P2 . Thus the
probability that all three vertices are in P2 (or in the direct product of the almost primes) is not (log N )−6
as one would expect, but roughly (log N )−4 , preventing the use of any measure of the form ν ⊗ ν.
In light of this our method is different, based on the hypergraph approach partly used already in [22], where
one reduces the problem to that of proving a hypergraph removal lemma for weighted uniform hypergraphs.
The natural approach is to use an appropriate form of the so-called transference principle [6], [16] to remove
the weights and apply the removal lemmas for “un-weighted” hypergraphs, obtained in [5], [15], [24]. This
way our argument also covers the main result of [3] and in particular of [8]. Recently another proof of
(one dimensional) Green-Tao theorem and the main result of [22], based on a removal lemma for uniform
hypergraphs, has been given in [1]. An interesting feature of the argument there is that it only uses (weaker
form) the so-called linear form condition of [8]. (Also in [2] uses only linear forms conditions.)
1
2
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
Recall that a set A ⊆ Pd has upper relative density α if
lim sup
N →∞
|A ∩ PdN |
=α
|PdN |
Let us state our main result.
Theorem 1.1. Let A ⊂ (PN )d with positive relative upper density α > 0 then A contains at least
N d+1
(affine copies) corners for some (computable) constant C(α).
C(α) (log
N )2d
As mentioned above, we will use the hypergraph approach which has been used to establish the the
existence of corners (and then that of general constellations) in dense subsets Zd [5] [15], first observed in
the simplest case in [19], where the key tool is the triangle removal lemma of Ruzsa and Szemerédi [17].
Theorem 1.2. (Triangles Removal Lemma [17]) If a graph of n vertices has at least δn2 edge-disjoint
triangles for some 0 < δ < 1. Then it in fact contains at least c(δ)n3 triangles for some c(δ) > 0.
In higher dimensions, there are hypergraph removal lemma (e.g.[15],[5], [24]) which follows from regularity lemma and counting lemma. In our weighted setting, this method allows us to distribute the weights
for primes (using the Green-Tao measure ν [8], see appendix) so that we can avoid dealing with higher moments of the Green-Tao measure ν. We will define the notion and prove some facts for independent weight
systems for which the weight systems related to corners is just a special case. The reason that we cannot
handle more general constellations is that we don’t quite have a suitable removal lemma (e.g. Thm. 5.1) for
general weight systems on non-uniform hypergraphs. Indeed for general constellations, our approach leads
to a weighted hypergraph with weights possibly attached to any lower dimensional hyperedge, making it
difficult to apply transference principle to remove weights.
The proofs of general multidimensional Szemeredi’s Theorem in the primes are given in [2], [25], [14]
using different method. The proof in [25] and [14] rely on “Infinite Linear Forms conditions” which rely on
Gowers Inverse Norm Theorem [11]. In this paper, we exhibit a method of Transference Principle to prove
a special case of corners. In particular we show how to prove weighted removal lemma (Theorem 5.3) from
unweight removal lemma (Theorem 5.1).
1.1. Notation. [N ] := {1, 2, ..., N }, [M, N ] := {M, M + 1, ..., M + N }, PN := P ∩ [N ].
d
Write x = (x1 , ..., xd ), y = (y1 , ..., yd ), ω ∈ {0, 1}d , let Pω : Z2d
N → ZN be the projection defined by
(
xj if ωj = 0
Pω (x, y) = u = (u1 , ..., ud ), uj =
yj if ωj = 1
For each I ⊆ [d], xI = (xi )i∈I . We denote the j th coordinate of xI by (xI )j . We may denote x for x[d]
when we work in ZdN . ωI means elements in {0, 1}|I| . Similarly we may write ω for ω[d] . We also define
PωI (xI , yI ) in the same way. ω|I is the ω restricted
Q to the index set I.
For finite sets Xj , j ∈ [d], I ⊆ [d] then XI := j∈I Xj and
(
Y
Xi , ωI (i) = 0
PωI (XI , YI ) =
Zi , Zi =
Yi , ωI (i) = 1
i∈I
If we want to fix on some position, we can write for example ω(0,[2,d]) means element in {0, 1}d such that
the first position is 0.
Also for each ω, define y1(ω) ∈ Zd by
(
0 if ωi = 0
(y1(ω) )i =
, 1 ≤ i ≤ d.
yi if ωi = 1
CORNERS IN DENSE SUBSETS OF Pd
(
1
(0(ω))i =
0
if
if
3
ωi = 0
, 1 ≤ i ≤ d.
ωi = 1
y0(ω) , 1(ω) ∈ Zd is also defined similarly.
For any finite set X and f : X → R, and for any measure µ on X,
Z
1 X
1 X
f dµ :=
Ex∈X f (x) :=
f (x),
f (x)µ(x)
|X|
|X|
X
x∈X
x∈X
Unless otherwise specified, the error term o(1) means a quantity that goes to 0 as N, W → ∞.
2. W EIGHTED HYPERGRAPHS AND BOX NORMS .
2.1. Hypergraph setting. First let us parameterize any affine copies of a corner as follow
Definition 2.1. A non-degenerate corner is given by the following set of d−tuples of size d + 1 in Zd (or
ZdN ):
{(x1 , ..., xd ), (x1 + s, x2 , ..., xd ), ..., (x1 , ..., xd−1 , xd + s), s 6= 0}
or equivalently,
{(x1 , ..., xd ), (z −
X
xj , x2 , ..., xd ), (x1 , z −
1≤j≤d
j6=1
with z 6=
X
xj , x3 , ..., xd ), ..., (x1 , ..., xd−1 , z −
1≤j≤d
j6=2
X
xj )}
1≤j≤d
j6=d
P
1≤i≤d xi
Now to a given set A ⊆ ZdN , we assign a (d + 1)− partite hypergraph GA as follows:
Let X1 = ... = Xd+1 := ZN be the vertex sets, and for j ∈ [1, d] let an element a ∈ Xj represent the
hyperplane xj = a, and an element a ∈ Xd+1 represent the hyperplane a = x1 + · · · + xd . We join these d
vertices (which represent d hyperplanes) if all of these d hyperplanes intersect in A. Then a simplex in GA
corresponds to a corner in A. Note that this includes trivial corners which consist of a single point.
For each I ⊆ [d + 1] let E(I) denote the set of hyperedges whose elements are exactly from vertices set
Vi , i ∈ I. In order to count corners in A, we will place some weights on some of these hyperedges that will
represent the coordinates of the corner. To be more precise we define the weights on 1−edges:
νj (a) = ν(a), a ∈ Xj , j ≤ d, νd+1 (a) = 1, a ∈ Xd+1 ,
and on d−hyperedges:
νI (a) = ν(ad+1 −
X
aj ), a ∈ E(I), |I| = d, d + 1 ∈ I
j∈I\{d+1}
ν[1,d] (a) = 1, a ∈ E([1, d])
We define measure spaces associated to our system of measure as follows. For 1 ≤ i ≤ d, let (Xi , dµXi ) =
(ZN , ν) and let µXd+1 be the normalized counting measure on Xd+1 = ZN .
In particular the weights are 1 or of the form νI (LI (xI )) where all linear forms {LI (xI )} are pairwise
linearly independent. This is an example of something we call independent weight system.
Definition 2.2 (Independent weight system). An independent weight system is a family of weights on the
edges of a d + 1−partite hypergraph such that for any I ⊆ [d + 1], |I| ≤ d, νI (xI ) is either 1 or of the form
QK(I)
j
j
j=1 ν(LI (xI )) where all distinct linear forms {LI }I⊆[d+1], 1≤j≤K(I) are pairwise linearly independent,
moreover the form LjI depends exactly on the variables xI = (xj )j∈I .
4
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
In fact for a weight system that arised from parametrizing affine copies of configurations in Zd , it is easy
to see from the construction that for any I ⊆ [d + 1], |I| = d all distinct linear forms {LkJ }J⊆I,1≤k≤K(J)
are linearly independent however we don’t need this fact in our paper. We define a measure on XI , I ⊆
[d + 1], |I| = d associated to an independent weight system by
Z
Y
f dµXI := ExI f I ·
νJ (xJ ),
XI
J⊆I,|J|<d
(that is we put only weights of order < d on hypergraph) as well as on X[d+1] by
Z
Y
νI (xI ),
f dµX[d+1] := Ex[d+1] f ·
X[d+1]
I⊆[d+1],|I|<d
and the associated multi-linear form, which will be used to estimate the numbers of prime configurations,
by
Z
Y
I
Λ(f , |I| = d) :=
f I dµX[d+1]
(2.1)
X[d+1] |I|=d
Note that we can also parameterize any configuration of the form {x, x + tv1 , . . . , x + tvd } in Pd using
an appropriate independent weight system. Now for each I = [d + 1]\{j}, 1 ≤ j ≤ d let
X
f I = 1A (x1 , ..., xj−1 , xd+1 −
xi , xj+1 , ..., xd ) · νI
1≤i≤d
i6=j
and for I = [d] let f I = 1A (x1 , ..., xd ). Hence we attach 1-weight to the hypergraph and d−weight to the
function. This is a way we distribute the weights in order to apply transference principle. As the coordinates
of a corner contained in Pd are given by 2d prime numbers, we have
Λ = Ex[d+1]
Y
|I|=d
fI
d
Y
ν(xi ) =
i=1
1
N d+1
2d
Y
X
ν(pi )
i=1
pi ∈A,1≤i≤2d
(pi )1≤i≤2d constitutes a corner
log2d N
|{number of corners in A}|
N d+1
(ignoring W-trick here and assuming that ν(N ) ≈ log N for now). Indeed if Λ ≥ C1 then
≈
|{number of corners in A}| ≥ C2
N d+1
.
log2d N
2.2. Basic Properties of Weighted Box Norm. In this section we describe the weighted version of Gowers’s uniformity norms on (d + 1)− partite hypergraph (called Box-norm) and the so-called Gowers’s inner
product associated to the hypergraph GA endowed with a weight system {νI }I⊆ [d+1],|I|≤d .
Definition 2.3. For each 1 ≤ j ≤ d, let Xj , Yj be finite set (in this paper we will define Xj = Yj := ZN )
with a weight system ν on X[d] × Y[d] . For f : X[d] → R, define
Z
Y
2d
kf kν :=
f (Pω[d] (x[d] , y[d] ))dµX[d] ×Y[d]
X[d] ×Y[d] ω
[d]
:= Ex[d] Ey[d]
Y
f (Pω[d] (x[d] , y[d] ))
ω[d]
Y Y
νI (PωI (xI , yI ))
|I|<d ωI
and define the corresponding Gowers’s inner product of 2d functions,
Z
D
E
Y
d
fω , ω ∈ {0, 1}
:=
fω[d] (Pω[d] (x[d] , y[d] ))dµX[d] ×Y[d]
ν
X[d] ×Y[d] ω
[d]
CORNERS IN DENSE SUBSETS OF Pd
:= Ex[d] Ey[d]
Y
5
Y Y
fω[d] (Pω[d] (x[d] , y[d] ))
ω[d]
νI (PωI (xI , yI ))
|I|<d ωI
d
So f, ω ∈ {0, 1}d ν = kf k2ν .
Definition 2.4 (Dual Function). For f, g : ZdN → R define the weight inner product
Z
Y
νI (xI ).
hf, giν :=
f · g dµX[d] = Ex∈Zd f (x)g(x)
N
X[d]
|I|<d
Define the dual function of f by
Df := Ey∈Zd
Y
N
Y Y
f (Pω (x, y))
νI (PωI (xI , yI ))
|I|<d ωI 6=0
ω6=0
So
d
kf k2ν
Y
= Ex∈Zd f (x)
N
Y Y
Y
νI (xI ) Ey∈Zd
f (Pω (x, y))
νI (PωI (xI , yI ))
N
|I|<d
ω6=0
|I|<d ωI 6=0
= hf, Df iν
It may not be clear immediately from the definition that k·kdν is a norm but this will follow from the
following theorem whose statements and the strategies of the proof are similar to analogue theorem for
unweight Gowers inner product.
Y
Theorem 2.1 (Gowers-Cauchy-Schwartz’s Inequality). | fω ; ω ∈ {0, 1}d | ≤
kfω kdν .
ω[d]
Proof. We will use Cauchy-Schwartz’s inequality and linear form condition. Write
1/2
Y Y
E
D
d
= Ex[2,d] ,y[2,d]
νI (PωI (xI , yI ))
fω ; ω ∈ {0, 1}
d
ν
|I|<d,1∈I
/ ωI
Ex1 ν(x1 )
Y
fω(0,[2,d]) (x1 , Pω[2,d] (x[2,d] , y[2,d] ))
ω[2,d]
ν{1}∪I (x1 , PωI (xI , yI ))
|I|<d−1,1∈I
/
1/2
Y
νI (PωI (xI , yI ))
Y
×
Y
|I|<d,1∈I
/ ωI
Y
Ey1 ν(y1 )
fω(1,[2,d]) (y1 , Pω[2,d] (x[2,d] , y[2,d] ))
ω[2,d]
Y
ν{1}∪I (y1 , PωI (xI , yI ))
|I|<d−1,1∈I
/
Applying the Cauchy Schwartz inequality in the x[2,d] , y[2,d] variables, one has
|hfω ; ω ∈ {0, 1}d idν |2 ≤ A · B
here,
Y
A = Ex[2,d] ,y[2,d]
Y
νI (PωI (xI , yI ))
|I|<d,1∈I
/ ωI
Y
× Ex1 ,y1 ν(x1 )ν(y1 )
fω(0,[2,d]) (x1 , Pω[2,d] (x[2,d] , y[2,d] ))fω(0,[2,d]) (y1 , Pω[2,d] (x[2,d] , y[2,d] ))
ω[2,d]
×
Y
Y
|I|<d−1,1∈I
/ ωI
ν{1}∪I (x1 , PωI (xI , yI ))ν{1}∪I (y1 , PωI (xI , yI ))
6
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
D
E
= fω(0) (Pω (x[d] , y[d] ))
dν
(0)
where fω̃ = f(0,ω̃∩[2,d]) for any ω̃[d] . And,
Y Y
B = Ex[2,d] ,y[2,d]
νI (PωI (xI , yI ))
|I|<d,1∈I
/ ωI
Y
fω(1,[2,d]) (x1 , Pω[2,d] (x[2,d] , y[2,d] ))fω(1,[2,d]) (y1 , Pω[2,d] (x[2,d] , y[2,d] ))
× Ex1 ,y1 ν(x1 )ν(y1 )
ω[2,d]
Y
×
Y
ν{1}∪I (x1 , PωI (xI , yI ))ν{1}∪I (y1 , PωI (xI , yI ))
|I|<d,1∈I
/ ωI
D
E
= fω(1) (Pω (x[d] , y[d] ))
dν
(1)
where fω̃ = f(1,ω̃∩[2,d]) for any ω̃[1,d] .
Then, apply Cauchy-Schwartz’s inequality in (x[3,d] , y[3,d] ) variables in the same way to end up with
Y
ω̃
|hfω ; ω ∈ {0, 1}d idν |4 ≤
hfω [1,2] ; ω ∈ {0, 1}d idν
ω̃[1,2] ∈{0,1}[1,2]
Iterate this, apply Cauchy-Schwartz’s inequality consecutively in (x[4,d] , y[4,d] ), ..., (x[d,d] , y[d,d] ) variables,
we end up with
Y
d
|hfω ; ω ∈ {0, 1}d idν |2 ≤
hf ω , ..., f ω idν , f ω = fω
ω[d]
≤
Y
d
kfω k2dν
ω[d]
Corollary 2.2. k·kdν is a norm for N is sufficiently large.
Proof. First we show nonnegativity. By the linear forms condition, k1kν = 1 + o(1). Hence by the
Gowers-Cauchy-Schwartz inequality, we have kf kdν & |hf, 1, ..., 1idν | ≥ 0 for all sufficiently large N .
Now
kf + gkdν = hf + g, ..., f + gidν
=
X
(
f
hhω1 , ..., hωd idν , hω =
g
d
ω∈{0,1}
≤
X
,ω = 0
,ω = 1
d
khω1 kdν ... khωd kdν = (kf kdν + kgkdν )2
ω∈{0,1}d
d
d
d
Also it follows directly from the definition that kλf k2dν = λ2 kf k2dν . Since the norm are nonnegative, we
have kλf kdν = |λ| kf kdν .
2.3. Weighted generalized von-Neumann inequality. The generalized von-Neumann inequality says that
the average Λ := Λd+1,ν (f I , I ⊆ [d + 1], |I| = d), see equation (2.1), is controlled by the weighted box
norm. We show this inequality in the general settings of an independent weight system.
CORNERS IN DENSE SUBSETS OF Pd
7
Theorem 2.3 (Weighted generalized von-Neumann inequality). Let I ⊆ [d + 1], |I| = d, f I : XI → R.
Let ν be an independent system of measure on X[d+1] that satisfies linear form conditions. Suppose f I are
dominated by ν i.e. |f I | ≤ νI . Write f (i) = f [d+1]\{i} , then
|Λd+1,ν (f (1) , ..., f (d+1) )| . min{kf (1) kdν , ..., kf (d+1) kdν }
Proof. We will apply Cauchy-Schwartz inequality and the linear forms condition. The idea is to consider
one of the variables say xj , as a dummy variable and write
Λ := Exj (...)Ex[d+1]\{j} (...)
then apply Cauchy Schwartz’s inequality to eliminate the lower complexity factors and use linear forms
condition to control the extra factor gained. We do this repeatedly d times. Each application of Cauchy
Schwartz’s inequality will cause a blow up in a variable then after successive application of CauchySchwartz’s inequality we can obtain an expression in the form of box norm.
First apply Cauchy-Schwartz’s inequality in xd+1 variable to eliminate f (d+1)
d
Y
Y
Y
(d+1)
(i)
|Λ| ≤ Ex[d] f
(x[d] )
νI (xI )Exd+1
f (x[d+1]\{i} )
νI (xI )
i=1
|I|<d,d+1∈I
/
1/2 νI (xI )
ν[d] (x[d] )
Y
≤ Ex[d] ν[d] (x[d] )
|I|<d,d+1∈I
/
|I|<d,d+1∈I
Y
|I|<d,d+1∈I
/
1/2 d
Y
νI (xI )
× Exd+1
f (i)
i=1
Y
|I|<d,d+1∈I
νI (xI )
Now by the linear forms condition on the face {X[d+1]\{d+1} }(as the linear forms defining an independent
weight system are pairwise linearly independent), we have
Y
Ex[d] ν[d] (x[d] )
νI (xI ) = 1 + o(1),
|I|<d,d+1∈I
/
hence
Y
|Λ|2 . Ex[d] ν[d] (x[d] )
νI (xI )
|I|<d,d+1∈I
/
× Exd+1 ,yd+1
d
Y
Y
f (i) (x[d]\{i} , Pωd+1 (xd+1 , yd+1 ))
i=1 ωd+1 ∈{0,1}
Y
Y
νI (xI\{d+1} , Pωd+1 (xd+1 , yd+1 ))
|I|<d ωd+1 ∈{0,1}
d+1∈I
Next we want to eliminate f (d) (x[d+1]\{d} ) ≤ ν[d+1]\{d} (x[d+1]\{d} ). Seperating the average in xd variable,
write
Y
Y
Y
|Λ|2 . Ex[d+1]\{d} ,yd+1
ν[d+1]\{d} (x[d−1] , Pωd+1 (xd+1 , yd+1 ))
νI (PωI∩{d+1} (xI , yI ))
|I|<d,d∈I
/ ω{d+1}∩I
ωd+1 ∈{0,1}
×
Y
νI (xI )Exd
d−1
Y
f (i) (x[d]\{i} , Pωd+1 (xd+1 , yd+1 ))
Y
νI (PωI∩{d+1} (xI , yI )) · ν[d] (x[d] )
|I|<d,d∈I ωI∩{d+1}
i=1
|I|<d
d,d+1∈I
/
Y
Again, by the linear forms condition on the face {Pωd+1 (X[d+1]\{d} , Y[d+1]\{d} )}ωd+1 ∈{0,1} , we have
Y
Y
Y
Y
Ex[d+1]\{d} ,yd+1
ν[d+1]\{d} (x[d−1] , Pωd+1 (xd+1 , yd+1 ))
νI (PωI∩{d+1} (xI , yI ))
νI (xI )
|I|<d,d∈I
/ ω{d+1}∩I
ωd+1 ∈{0,1}
is 1 + o(1) = O(1) and hence
|Λ|4 .Ex[d−2] ,xd ,yd ,xd+1 ,yd+1
Y
ω[d,d+1]
ν[d+1]\{d−1} (Pω[d,d+1] (x[d+1]\{d−1} , y[d+1]\{d−1} ))
|I|<d
d,d+1∈I
/
8
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
Y
×
Y
νI (Pω[d,d+1]∩I (xI , yI ))
|I|≤d,d−1∈I
/ ω[d,d+1]∩I
× Exd−1
d−2
Y
Y
Y
ν[d] (Pωd (x[d] , y[d] ))
νI (PωI∩[d,d+1] (xI , yI ))
|I|≤d ω[d,d+1]∩I
d−1∈I
i=1 ω[d,d+1]
×
Y
Y
f (i) (Pω[d,d+1] (xI , yI ))
Y
ν[d+1]\{d} (Pωd+1 (x[d+1]\{d} , y[d+1]\{d} ))
ωd+1
ωd
Continue using Cauchy-Schwartz inequality in xd−1 , ..., x2 in a similar fashion , using that
Y
Y
Y
Ex[d+1]\{r} ,y[d+1]\{r}
ν[d+1]\{r} (Pω[r+1,d+1] (x[d+1]\{r} , y[d+1]\{r} ))
νI (Pω[r+1,d+1]∩I (xI , yI ))
ω[r+1,d+1]
|I|≤d ω[r+1,d+1]∩I
r∈I
/
is 1+o(1) = O(1) by linear forms are on all faces {Pω[r+1,d+1] (X[d+1]\{r} , Y[d+1]\{r} )}ω[r+1,d+1] ∈{0,1}[r+1,d+1] .
Eventually, we obtain
Y
Y
Y
d
|Λ|2 . Ex[2,d+1] ,y[2,d+1]
f (1) (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
νI (Pω[2,d+1]∩I (xI , yI ))
ω[2,d+1]
|I|<d,1∈I
/ ω[2,d+1]∩I
× W (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
(2.2)
where
Y
W := W (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ); ω ∈ {0, 1}[2,d+1] ) := Ex1
Y
νI (x1 , Pω[2,d+1]∩I (xI\{1} , yI\{1} ))
|I|<d ω[2,d+1]∩I
×
d+1
Y
Y
ν[d+1]\{k} (Pω[2,d+1] (x[d+1]\{k} , y[d+1]\{k} ))
k=2 ω[2,d+1]\{k}
2d
Write the RHS of (2.2) = f (1) d + E
ν
where
Y
|E| ≤ Ex[2,d+1] ,y[2,d+1]
f (1) (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
ω[2,d+1]
Y
ω[2,d+1]
× Ex[2,d+1] ,y[2,d+1]
νI (PωI (xI , yI )) × |W − 1|
|I|<d,1∈I
/ ωI
We wish to show that E = o(1). Taking square,
Y
|E|2 ≤ Ex[2,d+1] ,y[2,d+1]
ν[2,d+1] (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
Y
Y
Y
Y
νI (PωI (xI , yI ))
|I|<d,1∈I
/ ωI
ν[2,d+1] (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
ω[2,d+1]
Y
Y
νI (PωI (xI , yI ))|W − 1|2
|I|<d,1∈I
/ ωI
The term on the first line is 1+o(1) by linear form condition on all the faces {Pω[2,d+1] (X[2,d+1] , Y[2,d+1] )}ω
.
[2,d+1]∈{0,1}[2,d+1]
So we just need to show
Y
Y Y
Ex[2,d+1] ,y[2,d+1]
ν[2,d+1] (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
νI (PωI (xI , yI ))W = 1 + o(1)
ω[2,d+1]
|I|<d,1∈I
/ ωI
(2.3)
Ex[2,d+1] ,y[2,d+1]
Y
ω[2,d+1]
ν[2,d+1] (Pω[2,d+1] (x[2,d+1] , y[2,d+1] ))
Y
Y
νI (PωI (xI , yI ))W 2 = 1 + o(1)
|I|<d,1∈I
/ ωI
(2.4)
(2.3) follows from linear form conditions on the faces X[d+1] × Y[2,d+1] . (2.4) follows from linear form
conditions on the faces X[d+1] × Y[d+1] and we are done.
CORNERS IN DENSE SUBSETS OF Pd
9
3. T HE DUAL FUNCTION ESTIMATE .
Functions with bounded dual norm are something that can be described as obstruction to Gowers uniformity. Gowers [5] demonstrates how to derive transference principle using the norm whose its dual is
an algebra norm. Dual of Gowers uniformity norm is not an algebra norm but by restricting to functions
dominated by a pseudorandom measure, its dual satisfies some nice algebraic properties that will be useful
to derive transference principle. One important property is the dual function condition stated in theorem 3.1
below.
In this section we prove the that dual norm finite product (say K terms) of dual function DF such that
F ≤ ν is bounded. Here we put a limit K(α) on the numbers of terms and obtain the uniform bound O(1)
(via Linear Forms Conditions) which is sufficient in our applications of hypergraph removal lemma. In [8]
does not have a restriction on the size of K but the bound is then not uniform , is of the form OK (1) (they
apply Correlation condition in order to avoid infinite Linear forms Condition ).
Theorem 3.1. For all K ≤ K(α) any independent measure system and any fixed J ⊆ [d + 1], |J| = d, let
F1 , ..., FK : XJ → R, Fj (xJ ) ≤ νJ (xJ ) be given functions. Then for each 1 ≤ K ≤ K(α) we have that
K
Y
∗
DFj d = O(1)
ν
j=1
Proof. We will denote by I the subsets of a fixed set J ⊆ [d + 1], |J| = d. First, write
Y Y
Y
Fj (Pω (x, y j ))
νI (PωI (xI , yIj ))
DFj (x) = Eyj ∈Zd
N
|I|<d ωI 6=0
ω6=0
Now assume kf kdν ≤ 1 then
K
K
Y
Y
Y
f,
DFj ν = Ex∈Zd f (x)
DFj (x)
νI (xI )
N
j=1
j=1
|I|<d
= Ex∈Zd f (x)Ey1 ,...yK ∈Zd
N
K Y
Y
N
j=1
ω6=0
j
Fj (Pω (x, y ))
Y Y
|I|<d
νI (PωI (xI , yIj ))
νI (xI )
ωI 6=0
We will compare this to the box norm to exploit the fact that kf kdν ≤ 1. To compare this to the Gowers’s
inner product, let us introduce the following change of variables:
For a fixed y ∈ ZdN , write y j 7→ y j + y for 1 ≤ j ≤ K then our expression takes the form
K
K Y
Y
Y Y Y
f,
DFj ν = Ey1 ,...,yK Ex f (x)
Fj (Pω (x, y + y j ))
νI (PωI (xI , yIj + yI )) νI (xI )
j=1
j=1
ω6=0
|I|<d ωI 6=0
Since ZdN is cyclic. This is equal to the average
K Y
Y
Y Y j
j
Ey1 ,...,yK ∈Zd Ex,y∈Zd f (x)
Fj (Pω (x, y + y ))
νI (PωI (xI , yI + yI )) νI (xI )
N
N
j=1
|I|<d ωI 6=0
ω6=0
For ω ∈ {0, 1}d , Y = (y 1 , . . . , y k ) ∈ (ZdN )k . We will define functions Gω,Y (x) : ZdN → R such that
f,
K
Y
j=1
DFj
ν
D
E
= Ey1 ,..,yK Gω,Y ; ω ∈ {0, 1}d
dν
10
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
To do this, let G0 (x) := f (x) and for each ω̃ 6= 0, Y , define
Y
1
K Y
Y
2d−|I|
− 1
j
j
Gω̃,Y (x) :=
Fj (x + y1(ω̃) )
νI ((x + y1(ω̃) ) I )
νI (xI ) 2d−|I|
×
j=1
|I|<d
|I|<d
Hence for ω̃ 6= 0
Y
K Y
Y
− 1
1
j 2d−|I|
j
νI (Pω̃ (x, y)I ) 2d−|I|
νI ((Pω̃ ((x, y+y ) I
×
Gω̃,Y (Pω̃ (x, y)) =
Fj (Pω̃ (x, y+y ))
j=1
|I|<d
|I|<d
Remark 3.1. For each I ⊆ [d] and fixed ωI , the number of ω[d] such that ω[d] |I = ωI is 2d−|I| and
Pω (x, y)|I = PωI (xI , yI ) ⇐⇒ ω|I = ωI
Hence
D
E
Gω,Y ; ω ∈ {0, 1}d
dν
= Ex,y∈Zd
N
×
Y
= Ex,y∈Zd
N
Y
Gω,Y (Pω (x, y)) ×
ω
Y Y
νI (PωI (xI , yI ))
|I|<d ωI
K YY
Y
1
j
j
d−|I|
2
Fj (Pω (x, y + y ))(
νI (Pω ((x, y) + y1(ω) ) I )
ω
j=1
|I|<d
νI (Pω (xI , yI )|I )
1
− d−|I|
2
×
Y Y
νI (PωI (xI , yI ))
|I|<d ωI
|I|<d
= Ex,y∈Zd
N
Y Y
Y
K Y
K Y
j
j
Fj (Pω (x, y + y )) ×
νI (PωI (xI , yI + yI )) νI (xI )
f (x)
j=1 ω6=0
j=1 ωI 6=0
|I|<d
Hence we have
hf,
K
Y
E
D
DFj iν = Ey1 ,..,yK Gω ; ω ∈ {0, 1}d
j=1
dν
Then by Gowers-Cauchy-Schwartz’s and arithmetic-geometric mean inequality, we have
K
Y
X
Y
d
hf,
DFj iν ≤ kf kdν
kGω,Y kd . 1 +
kGω,Y k2d
ν
j=1
ν
ω[d] 6=0
ω6=0
Hence to prove the dual function estimate, it is enough to show that
d
Ey1 ,...,yK kGω̃,Y k2d = O(1)
ν
For any fixed ω̃ 6= 0. Now
d
Ey1 ,...,yK kGω̃,Y k2d = Ey1 ,...,yK Ex,y
Y
ν
Gω̃,Y (Pω (x, y))
ω
Y Y
νI (PωI (xI , yI ))
|I|<d ωI
K Y
YY
1
j
j
≤ Ey1 ,...,yK Ex,y
ν[d] (Pω (x, y) + y1(ω̃)
)
νI ((Pω (x, y) + y1(ω̃)
)I ) 2d−|I|
ω j=1
×
Y
|I|<d
Y Y
− 1
νI (Pω (x, y)I ) 2d−|I| ×
νI (PωI (xI , yI ))
|I|<d ωI
|I|<d
= Ey1 ,...,yK Ex,y
K Y
Y
j=1
ω
ν[d] (Pω ((x, y) +
j
))
y1(ω̃)
I
Y Y
|I|<d ωI
νI (PωI (xI , yI ) +
j
)
y1(ω̃)
I
CORNERS IN DENSE SUBSETS OF Pd
11
by remark 1 above. As the linear forms appearing in the above expression are pairwise linearly independent
this is O(1) by the linear forms condition as required.
4. T RANSFERENCE P RINCIPLE
In this section, we will slightly modify the transference principle in [6] (see Theorem 4.6 below), which
will allow us to deduce results for functions dominated by a pseudo-random measure from the corresponding
result on bounded functions. The main difference between this paper and of [8] is that the dual function
may not be bounded. We will do this on the set on which our functions have bounded dual, and treat the
contributions of the remaining set as error terms. We will need the explicit description of the set Ω(T ) that
the dual function is bounded by T using the correlation condition (see appendix). In general, T will depend
on and T → ∞ as → 0 but when we apply removal lemma we will choose to be some small number
depending on α and hence if α is fixed then we can regard as a fixed small constant and T as a fixed large
constant.
We will work on functions f : XI → R, dominated
by νI . WLOG I = [d]. Let h·i be any inner product
R
on F := {f : X[d] → R} written as hf, gi = f · g dµ for some measure µ on X[d] .
4.1. Dual Boundedness on XI . One property of the dual functions that is used in [8] is their boundedness.
However in the weighted settings, this is generally not true. To get around this, we will be working on sets
on which the dual functions are bounded and treat the contributions of the remaining parts as error terms.
Consider any independent weight system. Let I ⊆ [d + 1], |I| = d, f : XI → R, |f | ≤ νI (WLOG
I = [d]). Recall
Y
Y Y
Df = Ey
f (Pω (x, y))
νI (PωI (xI , yI ))
|I|<d ωI 6=0
ω6=0
Write hωI =
LI (x)|0(ωI )
|Df | ≤
hence using correlation condition (see appendix), we have
Y
X
τ (W · (aωI1 hωI1 − aωI2 hωI2 ) + (aωI1 − aωI2 )b)
(4.1)
∅6=J⊆[d] (ωI1 ,ωI2 )∈TJ
where for each J ( [d], J 6= ∅
TJ := {{ωI1 , ωI2 }, ωI1 , ωI2 6= 0, ωI1 6= ωI2 , 1(ωI1 ) = 1(ωI2 ) = J : ∃c ∈ Q, LI1 (y1(ωI1 ) ) = cLI2 (y1(ωI2 ) )}
where aωIj ∈ Q are some constants (in our case of corners, they will be integers). Define
X
d
ΩJ (T ) = {(x[d] :
τ (W · (aωI1 hωI1 − aωI2 hωI2 ) + (aωI1 − aωI2 )b)) ≤ T 1/2 }
(4.2)
{ωI1 ,ωI2 }∈TJ
Ω(T ) =
\
ΩJ (T )
J([d]
So Df is bounded by T on Ω(T ) for any fixed T > 1.
4.2. Transference principle.
Definition 4.1. For each T > 1 we have th set Ω(T ) and define the following sets
F := {f : X[d] → R}
FT := {f ∈ F : supp(f ) ⊆ Ω(T )}
ST := {f ∈ FT : |f | ≤ ν[d] (x[d] ) + 2}
We define the following (basic anti-correlation) norm on FT
kf kBAC := max | hf, Dgi |
g∈ST
We have the following basic properties of this norm.
(4.3)
12
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
Proposition 1.
(1) g ∈ FT ⇒ Dg ∈ FT
(2) k·kBAC is a norm on FT and can be extended to be a seminorm on F. Furthermore, we have
kf kBAC = f · 1Ω(T ) BAC , f ∈ F.
(3) Span{Dg : g ∈ ST } = FT
P
P
(4) kf k∗BAC = inf{ ki=1 |λi |, f = ki=1 λi Dgi ; gi ∈ ST } for f ∈ FT
P
Remark 4.1. If f ∈
/ FT then supp(f ) * Ω(T ) so f is not of the form ki=1 λi Dgi ; gi ∈ FT as RHS is zero.
(1) Suppose (x̃1 , ..., x̃d ) ∈ Ω(T )C then there is an J ( [d] such that KJ (x̃[d]\J ) > T where KJ
is the function in the definition of ΩJ (T ) for some J. Let g ∈ FT then g(x̃[d]\J , xJ ) = 0 for all
xJ ∈ XJ So
Dg(x̃[d]\J , xJ ) = g(x̃[d]\J , xJ )E(x) = 0
for some function E so Dg ∈ FT .
(2) It follows directly from the definition that kf + gkBAC ≤ kf kBAC + kgkBAC and kλf kBAC =
|λ| kf kBAC for any λ ∈ R. Now suppose f ∈ FT , f is not identically zero then we need to show
that kf kBAC 6= 0. Since X and Z are finite sets, we have that kf k∞ = maxx,z |f (x, z)| < ∞.
Let g = γf where γ is a constant such that kgk∞ < 2 then g ∈ ST and hf, Dgi = hf, Dγf i =
d
γ 2 −1 hf, Df i > 0 so kf kBAC > 0
Now supp(Dg) ⊆ Ω(T ) we have for any f ∈ F
kf k
= sup | hf, Dgi | = sup | f · 1Ω(T ) , Dg | = f · 1Ω(T ) Proof.
BAC
BAC
g∈ST
g∈ST
(3) If there is an f ∈ FT , f is not identically zero and f ∈
/ span{Dg : g ∈ ST } So f ∈ span{Dg : g ∈
⊥
ST } then hf, Dgi = 0 for all g ∈ ST . So kf kBAC = 0 which is a contradiction.
P
P
(4) Define kf kD = inf{ ki=1 |λi | : f = ki=1 λi Dgi , gi ∈ ST } which can be easily verified to be a
P
norm on FT . Now let φ, f ∈ FT , f = ki=1 λi Dgi , gi ∈ ST , then
| hφ, f i | =
k
X
|λi || hφ, Dgi i | ≤ kφkBAC
i=1
k
X
|λi | ≤ kφkBAC kf kD
i=1
so
kf k∗BAC ≤ kf kD
Next for all g ∈ ST , we have kDgkD ≤ 1 then
kf kBAC = sup | hf, Dgi | ≤ sup | hf, hi | = kf k∗D
g∈ST
so kf kBAC ≤
kf k∗D
i.e.
kf k∗BAC
khkD ≤1
≥ kf kD . So kf k∗BAC = kf kD .
Now let us prove the following lemma whose proof relies on the dual function estimate. From here we
consider our inner product h·iν and the norm k·kν . This argument also works for any norm for which one
has the function has bounded dual norm.
Lemma 4.1. Let φ ∈ FT be such that kφk∗BAC ≤ C and η > 0. Let φ+ := max{0, φ}. Then there is a
polynomial P (u) = am um + ... + a1 u + a0 such that
(1) kP (φ) − φ+ k∞ ≤ η
(2) kP (φ)k∗dν ≤ RP,T (C)
P
j
+
where P (x) = m
j=0 aj x is a polynomial such that |P (x) − x | ≤ for all x ∈ [−CT, CT ] and RP,T is
the polynomial
m
X
RP,T (x) =
K|aj |xj
j=0
CORNERS IN DENSE SUBSETS OF Pd
13
here K is the constant in the dual function estimates, whuch may be taken to be something like 2 if N is
sufficiently large but let us just leave it as K for now.
Remark 4.2. Note that it is possible to choose P so that RP,T ( 1 ) = exp(( T )−O(1) ).
Proof. Suppose kφk∗BAC ≤ C then there exist g1 , .., gk ∈ ST and λ1 , ..., λk such that φ =
P
1≤i≤k |λi | ≤ C. Hence
|φ(x1 , ..., xd )| ≤ (
k
X
Pk
i=1 λi Dgi
and
|λi |)( max |Dgi (x1 , .., xd )|) ≤ CT
i=1
1≤i≤k
Hence the Range of φ = φ(Ω(T )) ⊆ [−CT, CT ]. Then by Weierstrass approximation theorem, there is a
polynomial P (which may depend on C, T, η) such that
|P (u) − u+ | ≤ η
and so kP (φ) − φ+ k∞ ≤ η and we have (1).
Now using the dual function estimate, we have
∗
X
j ∗
φ d ≤ (
λi Dgi )j d ≤
ν
ν
1≤i≤k
X
≤K
∀|u| ≤ CT
Hence kP (φ)k∗dν ≤
j=0 |am |KC
m
ν
1≤i1 ≤...≤ij ≤k
|λi1 ...λij | ≤ K(
1≤i1 ≤...≤ij ≤k
Pm
∗
|λi1 ...λij | Dgi1 ...Dgij d
X
X
|λi |)j ≤ KC j
1≤i≤k
≤ RP,T (C)
Now we are ready to prove the transference principle.
Theorem 4.2. Suppose ν is a pseudorandom independent weight system. Let f ∈ F and 0 ≤ f (x[d] ) ≤
ν[d] (x[d] ), let η > 0. Suppose N ≥ N (η, T ) is large enough, then there are functions g, h on X1 × ... × Xd
such that
(1) f = g + h on Ω(T )
(2) 0 ≤ g ≤ 2on Ω(T )
(3) h · 1Ω(T ) d ≤ ν
To prove this theorem,we have h · 1Ω(T ) ∈ ST and
d
kh · 1Ω(T ) kBAC ≥ hh · 1Ω(T ) , D(h · 1Ω(T ) )iν = kh · 1Ω(T ) k2d
ν
so it suffices to show
Theorem 4.3. With the same assumption in Theorem 4.2, there are functions g, h such that
(1) f = g + h on Ω(T )
(2) 0 ≤ g ≤ 2on Ω(T )
(3) h · 1Ω(T ) BAC ≤ Here the BAC-norm is the BAC-norm with respect to h·iν
The following lemma will be used in the next proof.
Lemma 4.4 (Hahn-Banach’s Theorem see e.g. corollary 3.2 in [6]). Let K1 , ..Kr be closed convex subsets
of Rd , each containing 0 and suppose f ∈ Rd cannot be written as a sum f1 + ... + fr , fi ∈ ci Ki , ci > 0.
Then there is a linear functional φ such that hf, φi > 1 and hg, φi ≤ c−1
i for all i ≤ r and all g ∈ Ki .
Proof of Theorem 4.3: Define
K := {g ∈ F : 0 ≤ g ≤ 2 on Ω(T )}
L := {h ∈ F : khkBAC ≤ }
14
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
Then it is clear that K, L are convex.(Also 0 ∈ K, 0 ∈ Int(L) and then 0 ∈ Int(K + L).) Assume that
f∈
/ K + L on Ω(T ) then by Lemma 4.4, there exists φ ∈ F such that
(1) φ, f · 1Ω(T ) ν > 1
(2) hφ, giν ≤ 1 ∀g ∈ K
(3) hφ, hiν ≤ 1 ∀h ∈ L
First, we claim that φ ∈ FT . To see this, suppose g is a function whose supp(g) ⊆ Ω(T )C i.e. g ≡ 0 on
C
Ω(T
) so g ∈ K. Since g ∈ K, hφ, giν ≤ 1 but g could be chosen arbitrarily on Ω(T ) so we must have
φ Ω(T )C ≡ 0 and hence φ ∈ FT . Now let
(
2 if φ(x[d] ) ≥ 0
g(x[d] ) =
0 otherwise
then g ∈ K and
hφ, giν = hφ+ , 2iν = 2 hφ+ , 1iν ≤ 1 ⇒ hφ+ , 1iν ≤
1
2
Now since φ ∈ FT , h ∈ L. Suppose kh · 1Ω(T )C kBAC ≤ 1 then we have
Hence if h0 ∈ FT and kh0 kBAC
hφ, h · 1Ω(T )C iν = hφ, hiν ≤ −1 .
≤ 1 then h0 · 1Ω(T ) BAC = kh0 kBAC ≤ 1 so
hφ, h0 iν ≤ −1 ∀h0 ∈ FT , h0 BAC ≤ 1
so kφk∗BAC ≤ −1 as k·kBAC is a norm on FT .
Now by the Lemma 4.1, there is a polynomial P such that
kP (φ) − φ+ k∞ ≤
1
8
and
kP (φ)k∗dν ≤ RP,T (C)
Then hP (φ), 1iν ≤ hP (φ) − φ+ , 1iν + hφ+ , 1iν ≤
1
2
+
1
8
Also, from the linear form condition, we have
d
kν[d] (x[d] ) − 1k2d = oN →∞ (1)
ν
so suppose N ≥ N (T, η) so that
d
kν[d] (x[d] ) − 1k2d ≤
ν
then
1
8RP,T (C)
1 1
∗ d≤1+1=3
=
hP
(φ),
1i
+
P
(φ),
ν
−
1
≤
+
+
kP
(φ)k
ν
−
1
d
[d]
[d]
ν
ν
ν
ν
ν
2 8
2 4
4
1 1 3
| ν[d] , φ+ ν | = | ν[d] , φ+ − P (φ) ν |+| ν[d] , P (φ) ν | ≤ kφ+ − P (φ)k∞ ν[d] , 1 ν + ν[d] , P (φ) ν ≤ · +
8 2 4
Hence
3
1
f · 1Ω(T ) , φ ν = hf, φiν ≤ hf, φ+ iν ≤ ν[d] , φ+ ν ≤ +
<1
4 10
which is a contradiction. Hence f ∈ K + L on Ω(T ).
P (φ), ν[d]
Now we can rephrase Theorem 4.3 as follow:
Theorem 4.5 (Transference Principle). Suppose ν is an independent weight system with kν − 1kdµ ≤ 0 =
exp(−( T1 )O(1) ). Let f ∈ F, 0 ≤ f ≤ ν and 0 < η < 1 T then there exists f1 , f2 , f3 ∈ F such that
(1) f = f1 + f2 + f3
CORNERS IN DENSE SUBSETS OF Pd
(2) 0 ≤ f1 ≤ 2, supp(f1 ) ⊆ Ω(T )
(3) kf2 kdν ≤ , supp(f2 ) ⊆ Ω(T )
(4) 0 ≤ f3 ≤ ν, supp(f3 ) ⊆ Ω(T )C , kf3 kL1ν .
15
1
T.
Proof. Let g, h be as in Theorem 4.3. Take f1 = g · 1ΩT , f2 = h · 1ΩT then f · 1ΩT = f1 + f2 . Let
f3 = f · 1ΩC . Now by linear form condition
T
kf3 kL1ν ≤
1
Ex f · Df ·
T [d]
Y
νI (xI )
I⊆[d],|I|<d
Y Y
Y
1
1
= Ex[d] Ey[d]
νI (xI )
νI ((PωI (xI , yI ))) .
T
T
I⊆[d] ωI 6=0
I⊆[d]
Remark 4.3. We will choose here to be as in the removal lemma (Theorem 5.3) and T = T () to be
δ()−1 for the δ() in the removal lemma.
5. R ELATIVE H YPERGRAPH R EMOVAL L EMMA
First let us recall the statement of a version of functional hypergraph removal lemma [24].1 Recall the
definition of Λ in equation (2.1).
Theorem 5.1. Given finite measure spaces (X1 , µX1 ), ..., (Xd+1 , µXd+1 ) and f (i) : XI → [0, 1], I =
[d + 1]\{i} Let > 0, suppose |Λd+1 (f (1) , ..., f (d) , f (d+1) )| ≤ . Then for 1 ≤ i ≤ d, there exists
Ei ⊆ X[d+1]\{i}
such that
Q
1≤j≤d+1 1Ej
≡ 0 and for 1 ≤ i ≤ d + 1,
Z
Z
···
f (i) · 1E C dµX1 · · · dµXd dµXd+1 ≤ δ()
X1
Xd+1
i
where δ() → 0 as → 0.
The proof of removal lemma relies on functional version of Szemerédi’s Regularity Lemma [24]. If B is
a finite factor of X i.e. a finite σ−algebra of measurable sets in X, then B is a partition of X into atoms
A1 , ..., AM . Let f : X → R be measurable
then we define the conditional expectation E(f |B) : X → R is
R
defined by E(f |B)(x) = (1/|Ai |) Ai f (x)dµX if x ∈ Ai (defined up to set of measure zero). We say that
B has complexity at most m if it is generated by at most m sets. If BX is a finite factor of X with atoms
A1 , ..., AM and BY is a finite factor of Y with atoms B1 , ..., BN then BX ∨ BY is a finite factor of X × Y
with atoms Ai × Bj , 1 ≤ i ≤ M, 1 ≤ j ≤ N.
Theorem 5.2 (Szemerédi’s Regularity Lemma [24]2). Let f : X[d] → [0, 1] be measurable, let τ > 0
and F : N → N be arbitrary increasing functions (possibly depends on τ ). Then there is an integer
M = OF,τ (1), factors BI (I ⊆ [d], |I| = d − 1) on XI of complexity at most M such that f = f1 + f2 + f3
where
W
• f1 = E(f | I⊆[d],|I|=d−1 BI ).
• kf2 kL2ν ≤ τ.
1In fact the paper [24] proves this theorem only with the counting measure (with thenotion of e−discrepancy in place of Box
norm). But the proof also works for any finite measure that has direct product structure (with the notion of weighted Box Norm).(see
[21] for the case of probability measures in d = 2, 3). However we don’t know how to genralize this argument to arbitrary measure
on the product space. If we can prove this theorem for any measure µX1 ×...×Xd then we would be able to prove multidimensional
Green-Tao’s Theorem.
2This theorem is proved for counting measure in [24] but the proof would work for any product measure on the product spaces.
16
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
• kf3 kdν ≤ F (M )−1 .
• f1 , f1 + f2 ∈ [0, 1].
Remark 5.1. A consequence
from this lemma that we will use later is the following: since f1 is a constant
W
on each atom of I.|I|=d−1 BI , we can decompose f1 as a finite sum of O(M ) = OF,τ (1) terms of lower
Q
complexity functions i.e.a finite sum of product di=1 Ji where Ji is a function in x[d]\{i} variable and takes
values in [0, 1]. One advantage of exploting conditional expectation is that it is easy to see that f1 is positive.
Our main goal is to prove the following version of weighted removal lemma using Theorem 5.1 and
transference principle.
(i)
Theorem 5.3 (Weighted Simplex-Removal Lemma).
Q Suppose 0 ≤ f (x[d+1]\{i} ) ≤ ν[d+1]\{i} (x[d+1]\{i} ).
Let > 0, Suppose |Λ| ≤ then there exist Ei ⊆ j∈[d+1]\{i} Xj such that for 1 ≤ i ≤ d + 1,
Y
1Ei ≡ 0
•
Ri∈[d+1] R
• X1 · · · Xd+1 f (i) 1E C dµX1 · · · dµXd+1 ≤ δ()
i
where δ() → 0 as → 0.
Proof. Using the transference principle (Theorem 4.6) for 1 ≤ i ≤ d + 1, write f (i) = g (i) + h(i) + k (i)
where
(1) f (i) = g (i) + h(i) + k (i)
(2) 0 ≤ g(i) ≤ 2, supp(g (i) ) ⊆ Ω(i) (T )
(3) h(i) d ≤ , supp(h(i) ) ⊆ Ω(i) (T )
ν
(4) k (i) = f (i) · 1(Ω(i) )C (T )
where
Ω(i) (T ) = {x[d+1]\{i} : |Df (i) | ≤ T }, 1 ≤ i ≤ d
Step 1: We’ll show that if T ≥ T () is sufficiently large then
Λd+1 (g (1) + h(1) , ..., g (d+1) + h(d+1) ) = Λd+1 (f (1) − k (1) , ..., f (d+1) − k (d+1) ) . .
Proof of Step 1: For I ⊆ [d + 1], the term on LHS can be written as a sum of the following terms:
(
−k (i) if i ∈ I
Λd+1,I (e(1) , ..., e(d) , e(d+1) ), e(i) =
f (i)
if i ∈
/I
6 ∅ then
If I = ∅ then Λd+1 (f (1) , ..., f (d) , f (d+1) ) ≤ by the assumption. Suppose I = {i1 , ..., ir } =
Z
Z
Y
(1)
(d) (d+1)
(1)
(d+1)
···
f ···f
·
1(Ω(i) )C dµX1 · · · dµXd+1 |Λd+1,I (e , ..., e , f
)| = X1
≤ Ex[d+1]
Xd+1
i∈I
Y
νI (xI )1(Ω(i1 ) )C
I⊆[d+1],|I|≤d
≤
.
1
Ex Ey
T d+1 [d+1]\{i1 }
Y
νI (xI )
I⊆[d+1],|I|≤d
Y
ωI 6=0
I⊆[d+1]\{i1 }
1
T
by linear form condition.
Step 2 We’ll show Λd+1 (g (1) , ..., g (d+1) ) . if N ≥ N ().
Proof of step 2: Write g (i) = g (i) + h(i) − h(i) = f (i) · 1Ω(i) (T ) − h(i) then we have
0 ≤ f (i) · 1Ω(i) (T ) ≤ νi , kh(i) kdν ≤ νI (PωI (xI , yI ))
CORNERS IN DENSE SUBSETS OF Pd
17
so by the weighted von-Neumann inequality and step 1 , we have
|Λd+1 (g (1) , ..., g (d+1) )| = |Λd+1 (g (1) + h(1) , ..., g (d+1) + h(d+1) ) − Λd+1 (h(1) , .., h(d) , h(d+1) )|
. + oN →∞ (1)
.
if τ ≤ , N ≥ N () and the proof of step 2 is completed.
Now since 0 ≤ g (i) ≤ 2 then (after normalizing) using the unweight hypergraph removal lemma (Theorem
5.1), we have
Y
F(i) ⊆ X[d+1]\{i} , F(i) ∈ B[d+1]\{i} , compl(B[d+1]\{i} ) ≤ M such that
1Fk ≡ 0 and
1≤k≤d+1
Z
Z
···
X1
Xd+1
g (i) · 1F C dµX1 · · · dµXd +1 . δ()
i
so
Z
Z
f (i) · 1F C dµX1 · · · dµXd+1 . δ() +
···
X1
Z
i
Xd+1
Z
···
X1
Xd+1
h(i) · 1F C dµX1 · · · dµXd dµXd+1 +
i
|
Z
{z
Z
···
+
X1
}
(A)
Xd+1
f (i) · 1ΩC (T ) 1F C dµX1 · · · dµXd+1
i
i
{z
|
(B)
}
Now for our purpose, it suffices to show (A), (B) . .
Estimate for (A): By the regularity lemma 3, the function 1F C could be written as a sum of OF,τ (1)
i
Q
(i)
(i)
of functions of the form j∈[d+1]\{i} uj where uj is a [0, 1]- valued function in x[d+1]\{i,j} . Hence it
Q
(i)
suffices to consider functions of the form functions of the form j∈[d+1]\{i} uj in place of 1F C . We have
i
the following easy lemma which roughly states that Gowers uniform functions are orthogonal to lower order
functions (in particular, it is uniformly distributed across lower order sets.)
(i)
Lemma 5.4. For h(i) : X[d+1]\{i} → R and uj be as above. Then
Z
2d
Z Z
Y
(i)
(i)
···
h
uj dµXd+1 dµX1 · · · dµXd dµXd+1
≤ khkdµ
X1
Xd
Xd+1
1≤j≤d+1
j6=i
Proof of Lemma. WLOG assume i < d + 1. Applying Cauchy-Schwartz’s inequality in (x1 , . . . xd )− vari(i)
able and using that ud+1 is bounded, we have
Z
2 2d−1
Z Z
Y (i)
(i)
i
···
h
uj dµXd+1 ud+1 dµX1 · · · dµXd dµXd+1
X1
Xd
Z
Z
Z
≤
Xd+1
···
X1
×
Z
···
X1
Y
(i)
h
Xd
Z
1≤j≤d
j6=i
Xd+1
(i)
uj dµXd+1
2
dµX1 · · · dµXd
1≤j≤d
j6=i
2
uid+1 dµX1
2d−1
· · · dµXd
Xd
3We need this since we don’t have something like kf gk
ν
≤ kf kν kgkν
18
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
Z
Z
Z
Z
h(i) (x[d+1]\{i} , xd+1 )h(i) (x[d]\{i} , yd+1 )
···
.
Y
Yd+1
Xd+1
Xd
X1
(i)
(i)
uj (x[d]\{i} , xd+1 )uj (x[d]\{i} , yd+1 )dµX1
2d−1
· · · dµXd+1 dµYd+1
1≤j≤d
j6=i
(i)
Continue apply Cauchy-Schwartz’s inequality in this way d − 1 more times to eliminate each uj we end
up with
Z Z Y
d
h(i) (PωI (xI , yI ))dµXI dµYI = khk2d
XI
µ
YI ω
I
as required
Now we apply the lemma we can estimate the expression (A):
Z
2d
Z Z
(i)
···
h · 1F C dµX1 · · · dµXd dµXd+1
i
X1
Xd
Xd+1
Z
Z
Z
(i)
···
.
X1
h
Xd
Xd+1
(i)
uj dµXd+1
Y
2d
dµX1 · · · dµXd dµXd+1
1≤j≤d+1
j6=i
d
≤ khk2d ≤ µ
as required.
Estimate for (B) : Ignoring 1F C term, we have
i
Z
Z
(i)
···
f · 1(Ω(i) (T ))C · 1F C dµX1 · · · dµXd+1 i
X1
Xd+1
Z
Z
≤
···
(ν[d+1]\{i} ) · 1(Ω(i) (T ))C dµX1 · · · dµXd+1
X1
≤
Xd+1
1
Ex
Ey
ν
(x
)
T [d+1]\{i} [d+1]\{i} [d+1]\{i} [d+1]\{i}
Y
|I|≤d,i∈I
/
Y
νI (xI )
νI (PωI (xI , yI )) .
ωI 6=0
I⊆[d+1]\{i}
1
,
T
by the linear forms condition. Hence if we choose sufficiently large T e.g. T = δ()−1 then
Z
Z
···
f (i) · 1F C dµX1 · · · dµXd+1 . δ().
X1
Xd+1
i
6. PROOF OF THE MAIN RESULT
)
6.1. From ZN to Z. Now recall that νδ1 ,δ2 (n) ≈ φ(W
W log N, δ1 N ≤ n ≤ δ2 N, δ1 , δ2 ∈ (0, 1] for a
d
sufficiently large prime N in the residue class b (mod W ). By pigeonhole principle choose a b ∈ (Z×
W)
such that
Nd
|A ∩ (W Z)d + b| ≥ α
(logd N )φ(W )d
Now consider Ab = {n ∈ [1, N/W ]d : W n + b ∈ A} and let δ2 ∈ (0, 1) then by the Prime Number
N
Theorem there is a prime N 0 such that δ2 N 0 = (1 + δ) W
for arbitrarily small real number δ. Then if N is
CORNERS IN DENSE SUBSETS OF Pd
19
sufficiently large and δ is sufficiently small with respect to α then
αδ2d
(N 0 W )d
2 (logd N 0 )φ(W )d
|Ab ∩ [1, δ2 N 0 ]d | ≥
(6.1)
On the other hand by Dirichlet’s theorem on primes in arithmetic progressions, the number n ∈ [1, N 0 ]d \[δ1 N 0 , N 0 ]d
0 W )d
for which W n + b ∈ Pd is ≤ cd 1 log(N
Hence the estimate (6.1) holds for A0 := AW ∩ [δ1 N 0 , δ2 N 0 ]d
d
N 0 φ(W )d
as well provided that δ1 is small enough.
Now we may consider A0 in place of A (we are working in the group ZdN 0 instead). Now if we identify
0
0
the group ZdN 0 with [− N2 , N2 ]d then for a sufficiently small δ1 , δ2 any points in A0 are the same when we
0
0
change from ZdN 0 to [− N2 , N2 ]d (no wrap around issue).
6.2. Proof of the Main Theorem. To prove the theorem, suppose on the contrary that A0 contains less than
N 0d+1
corners.( = c(α)) then
(log
N 0 )2d
Λd+1 (f (1) , ..., f (d+1) )
X Y
X
1A0 (x1 , ..., xi−1 , xd+1 −
= (N 0 )−(d+1)
xj , xi+1 , ..., xd )νI 1A0 (x1 , ..., xd ) · ν(x1 )...ν(xd )
x[d+1] 1≤i≤d
≤
.
1
N 0d+1
1
X
1≤j≤d
j6=i
Y
1A0 (p1 , ..., pk−1 , pd+k , pk+1 , ..., pd )1A (p1 , ..., pd )ν(p1 )...ν(p2d )
1≤k≤d
pi ∈A0 ,1≤i≤2d
that consitutes a corner
0 2d
N 0d+1
φ(W ) log N
W
× (The number of corners in A0 )
≤
Now assume that Λd+1 (f (1) , ..., f (d) , f (d+1) ) . then by the relative hypergraph removal lemma
∃Ei , 1 ≤ i ≤ d + 1, Ei ⊆ X[d+1]\{i} := X̃i ,
such that
Y
Z
1Ei ≡ 0,
X̃i
1≤i≤d+1
f (i) 1E C dµX̃i . δ()
i
P
where δ() → 0 as → 0. Let A0 = A ∩ [δ1 N, δ2 N ]d , z = 1≤j≤d xj , gA0 := g · 1A0 for any function g
then
X
(1)
(2)
(d)
(d+1)
Λ̃ := N 0−d
fA0 (x2 , ..., xd , z)fA0 (x1 , x3 , ..., xd , z)...fA0 (x1 , x2 , ..., xd−1 , z)fA0 (x1 , ..., xd )
(x1 ,...,xd )∈A0
X
≥ N 0−d
ν(x1 )...ν(xd )
(x1 ,...,xd )∈A0
& (N 0 )−d
d
α · (N 0 W )d
φ(W )
= α.
log N 0 ·
W
(φ(W ) log N 0 )d
for arbitrarily large N 0 . Now
(1)
(1)
(d+1)
Λ̃ = Ex[d] (fA0 1E1 + fA0 1E C )...(fA0
1
(1)
(d+1)
Now we have by the assumption Ex[d] fA0 · 1E1 ...fA0
term individually.
(d+1)
1Ed +1 + fA0
1E C )
d+1
· 1Ed+1 ≡ 0 so we just need to estimate each other
20
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
(d+1)
(2)
(1)
· 1E ± , where F ± can be either F or F C for any set F . Now
Consider Ex[d] fA0 · 1E C fA0 · 1E ± ...fA0
1
2
d+1
since
(d+1)
(j)
≤1
0 ≤ fA0 1E ± ≤ ν(xj ), d ≥ j ≥ 2 and 0 ≤ fA
j
We have
(d+1)
(1)
(2)
Ex[d] fA0 ·1E C fA0 ·1E ± ...fA0 ·1E ±
1
1
d+1
≤
(1)
Ex[d] fA0 ·1E C ν(x2 )...ν(xd )
1
Z
=
X̃1
f (1) ·1E C dµX2 · · · dµXd+1 . δ().
1
In the same way, we have for any 1 ≤ i ≤ d + 1,
(i)
Ex[d] fA0 · 1E C
Y
(f (j) · 1E ± ) . δ()
i
j
1≤j≤d+1,j6=i
So if N 0 > N (α) then
(1)
(2)
(d)
(d+1)
Ex[d] fA0 (x2 , ..., xd , u)fA0 (x1 , x3 , ..., xd , u)...fA0 (x1 , ..., xd−1 , u)fA0
(x1 , ..., xd ) . δ() = o(α)
0d+1
N
This is a contradiction. Hence there are & (log
corners in A. Note that the number of degenerated
N 0 )2d
0d
corners is at most O( (logNN 0 )d ) as the corner is degenerated (and will be degenerated into a single point ) iff
P
(N 0 )d+1
z = 1≤j≤d xj . Hence there are at least c(α) (log
corners.
N 0 )2d
Remark 6.1. It possible to extract and explicit bound c(α) from this argument but it is not a good bound
(iterated tower type) due to the use of regularity lemma. See [7]. This is also the best bound in integers case.
For d = 2, the best bound is due to Shkredov [18] which is an exponents bound.
A PPENDIX A. G REEN -TAO ’ S M EASURE AND P SEUDORANDOMNESS
A.1. Pseudorandom Measure Majorizing Primes. In analytic number theory, the following Mangoldt
function is used as a characteristic function on primes.
(
log p if n = pk , k ≥ 1
Λ(n) =
0
otherwise
Primes has local obstructions that prevents it from being random : Λ(n) is concentrated on just φ(q) residue
classes (mod q). The small primes will cause this kind of effect more than the larger primes as they have
larger density. However, we can get rid of this kind of obstruction on all small residue classes and will not
affect much what we are Q
counting by the so called W-Trick[8]: Let ω(N ) be a sufficiently slowly growing
function of N . Let W = p≤ω(N ) p.Let b be any positive integer with (b, W ) = 1, so by the Prime Number
Theorem, we have W = exp((1 + o(1))ω(N )) and we have that PW,b is uniformly distributed (mod q)
for q ≤ W .
Remark A.1. We have to choose W to grow sufficiently slow in N , ω(N ) = log log N is enough. If we let
W grow with N then the error term from linear form and correlation conditions would go to 0 an N → ∞.
It turns out that we can choose W to be arbitrarily large fixed constant (see also remark in Sec 11 in [8]
or overspill principle in [23]) and in this case for error term to go to 0 we need to let both N, W → ∞.
Keeping W independent of N may be important to extgract the quantitative bound of the main theorem.
We look at PN,W,b = {n : W n + b ∈ PN } in place of PN and for A ⊆ P, we look at {n : W n + b ∈
A ∩ [N ]} instead . We do this by identifying n with the original W n + b. Applying W-trick doesn’t affect
much what we are counting.4
4Recall that Q
p≤x (1
− p1 )−1 = eC0 log x + O(1) hence
≈
1
N
φ(W ) log W N
N/W
≈
W
φ(W )
= eC0 log x + O(1) The density of PW n+b in ZW n+b is
eC0 log ω(N ) + O(1)
W
1
(if W N ) ≈
φ(W ) log N
log N
CORNERS IN DENSE SUBSETS OF Pd
21
We also define the modified von-Mangoldt function by
Definition A.1 (Modified von-Mangoldt function). For any fixed (b, W ) = 1,
(
φ(W )
W log(W n + b) if W n + b is prime.
Λb (n) =
0
otherwise.
Modified Mangoldt function in dimension d is defined to be d−fold tensor product of Λb that is
d
Λb (x1 , ..., xd ) = Λb1 (x1 ) · · · Λbd (xd ), b = (b1 , ..., bd ) ∈ Z×d
N
Remark A.2. Note that the modified Mangoldt function is not supported on higher prime powers. From the
Prime Number Theorem in Arithmetic Progression, we have En≤N Λb (n) ∼ 1. If W is sufficiently large then
we don’t have much effect from local obstructions and the function Λb is more pseudorandom. For example,
it could be shown ([10])that
kΛ − 1kU s+1 [N ] = o(1)
which is not true for Λ.
Now we recall the definition of Green-Tao measure and the definiton of pseudorandomness measure
according to [8].
Definition A.2 (Goldston-Yildirim sum). [12],[8]
ΛR (n) =
X
d|n,d≤R
We may take R = N
d−1 2−d−5
µ(d) log
R
d
Now we define the Green-Tao measure:
Definition A.3 (Green-Tao’s pseudorandom measure). [8] For given small parameters 1 ≥ δ1 , δ2 > 0 ,
define a function νδ1 ,δ2 : ZN → R
(
φ(W ) ΛR (W n+b)2
if δ1 N ≤ n ≤ δ2 N
W
log R
νδ1 ,δ2 (n) = ν(n) =
0
otherwise
Now we summarize some properties of ν that will be used
• ν satisfies linear forms for any parameters depending d or α.
• ν(n) ≥ d−1 2−d−6 Λb (n). To see this, we may assume that W n+b is prime then δ1 N > R, ΛR (W n+
1) = log R = d−1 2−d−5 log N so we have our claim provided ω(N ) is sufficiently slow growing in
N . Moreover, if N is a sufficiently large prime in residue class b (mod W ) and is in the support of
)
ν then ν(N ) ≈ φ(W
W log N .
• in dimension d we define Green-Tao measure to be d-fold tensor product of ν i.e. ⊗d ν(x1 , ..., xd ) =
ν(x1 )...ν(xd ).
Definition A.4 (Linear Form Condition). Let m0 , t0 ∈ N be parameters then we say that ν satisfies
(m0 , t0 )− linear form condition if for any m ≤ m0 , t ≤ t0 , suppose {aij }1≤i≤m are subsets of inte1≤j≤t
P
gers and bi ∈ ZN . Given m (affine) linear forms Li : ZtN → ZN with Li (x) = 1≤j≤t aij xj + bi for
1 ≤ i ≤ m be such that each φi is nonzero and they are pairwise linearly independent over rational. Then
Y
E(
ν(Li (x)) : x ∈ ZtN ) = 1 + oN →∞,m0 ,t0 (1)
1≤i≤m
Definition A.5 (Correlation Condition). We say that a measure ν satisfies (m0 , m1 , ..., ml2 )− correlation
condition if there is a function τ : ZN → R+ such that
(1) E(τ (x)m : x ∈ ZN ) = Om (1) for any m ∈ Z+
22
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
(2) Suppose
• φi , ψ (k) : ZtN → ZN (1 ≤ i ≤ l1 , 1 ≤ k ≤ l2 , l1 + l2 ≤ m0 ) are all pairwise linearly
independent (over Z) linear forms
(g)
(g) (g)
• For each 1 ≤ g ≤ l2 , 1 ≤ j < j 0 ≤ mg we have agj 6= 0, and aj ψ (g) (x) + hj , aj 0 ψ (g) (x) +
(g)
hj 0 are different (affine) linear forms.
then, we have
Ex∈Z d
l1
Y
N
ν(φk (x))
(k)
(k)
ν(aj ψ (k) (x)+hj )
Q
≤
l2
Y
(k) (k)
(k) (k)
(k)
(k)
τ W (aj 0 hj −aj hj 0 )+(aj 0 −aj )b
X
k=1 1≤j<j 0 ≤mk
k=1 j=1
k=1
where W =
mk
l2 Y
Y
p≤ω(N ) p.
Theorem A.1. The green-Tao measure ν satisfies linear forms and correlation conditions on any parameters
that may depend on d or α (not in N ).
Proof. The proof for linear form condition is the same as in [8], the correlation condition is very slight
different and can be proved in the same manner. We present a sketch of a proof of the correlation condition
−1 −d−5
here. Let B be a box of size R10M where M = m0 ...ml2 , R = N d 2
. It suffices to proof the following
(see Theorem 9.6 in [8])
Y
2 Y
2
mk
m0
l2 Y
(k) (k)
(k)
Ex∈B
ΛR (W φk (x) + b)
ΛR W · (aj ψ (x) + hj ) + b
k=1 j=1
k=1
≤ CM
W log R
φ(W )
M Y
l2 Y 1 + OM (p
− 12
)
(A.1)
k=1 p|∆k
Where for each k ≥ 1
∆k :=
Y
(k)
(k)
(k) (k)
(k) (k)
W · (aj 0 hj − aj hj 0 ) + (aj 0 − aj )b
1≤j<j 0 ≤mk
S 0
k
Define Mj := m0 +m1 +...+mj . Write [M ] = m
j=1 Ij ∪∪j=1 Imj , Ij := {j}, j ≤ m0 , Imj = (Mj−1 , Mj ]
(
φi
if i ∈ Ij , j ≤ m0
(k)
Define ψi :=
(k)
ψ , if i ∈ Ik , k > m0
Now for each i ∈ Ik , k ≥ m0 , write
(k)
(k)
(k)
θi (x) := W · ai−Mk−1 ψi−Mk−1 + hi−Mk−1 + b
and for each i ∈ Ij , j ≤ m0 , define
(i)
(i)
θi (x) := W φi (x) + b (i.e. ai = 1, hi = 0, ψ i = φi )
For each X ⊆ [M ], define
ωX (p) = Ex∈Zd
Y
N
1θi (x)≡0
(mod p)
i∈X
0 ) then as in [8] we can write LHS of (A.1) as the following integral
Write z = (z1 , ..., zM ), z 0 = (z10 , .., zM
plus a small error term
Z
Z
Y Rzj +zj0
(2πi)−M
···
F (z, z 0 )
dzj dzj0
(A.2)
2z02
z
Γ1
Γ1
j
j
1≤j≤M
| {z }
2M
CORNERS IN DENSE SUBSETS OF Pd
where Γ1 is the line <(z) = σ > 0, F (z, z 0 ) :=
Q
p Ep (z, z
0 ),
23
where
0
Ep (z, z 0 ) :=
(−1)|X|+|X | ωX∪X 0 (p)
X
X,X 0 ⊆[M ]
p
P
j∈X
P
zj + j∈X zj0
To ensure the convergence we have the following estimate
Lemma A.2 (Local factor estimate). Let X ⊆ [M ]
(1) If X = ∅ then ωX (p) = 1
(2) If p ≤ ω(N ), X 6= ∅ then ωX (p) = 0
(
= p−1 if |X| = 1
(3) If p > ω(N ) and X ⊆ Ik is nonempty then ωX (p)
Furthermore, if |X| > 1
≤ p−1 if |X| > 1
and p - ∆k then ωX (p) = 0.
(4) If p > ω(N ) and there are k1 6= k2 such that both X ∩Ik1 , X ∩Ik2 are nonempty then ωX (p) ≤ p−2 .
Proof of the Lemma.
(1) The first statement is trivial.
(k) (k)
(k)
(2) If p ≤ ω(N ), j ∈ X then W · (aj ψj + hj ) + b ≡ b (mod p) which gives the result since
(b, p) = 1.
(3) Suppose p > ω(N ) Firstly, if X ⊆ Ik , |X| = 1 then
Ex∈Zd 1W ·(a(k)
= p−1
(k)
(k)
ψ
(x)+h
)+b≡0 (mod p)
N
i−Mk−1
Now if |X| > 1 with
i−Mk−1
i−Mk−1
j, j 0
ωX (p) ≤ Ex∈Zd
N
∈ X then using the previous estimate, we have
1W ·(a(k)
= p−1 .
(k)
(k)
ψ
(x)+h
)+b≡0 (mod p)
j−Mk−1
j−Mk−1
j−Mk−1
Now assume p - ∆k we use the estimate
ωX (p) ≤ Ex∈Zd 1W ·(a(k)
(k)
(k)
ψ
(x)+h
N
j−Mk−1
(k)
j−Mk−1 )+b≡0
j−Mk−1
(k)
(k)
(mod p)
(k)
·1W ·(a(k)
(k)
(k)
j−Mk−1 ψj−Mk−1 (x)+hj−Mk−1 )+b≡0
(k)
(mod p)
(k)
Now if p|W · (aj ψj (x) + hj ) + b, p|W · (aj ψj (x) + hj ) + b then
(k) (k)
(k) (k)
(k)
(k)
p|W · (aj 0 hj − aj hj 0 ) + (aj 0 − aj )b
so p|∆k . Hence if p - ∆k then ωX (p) = 0.
(4) Assume j ∈ X ∩ Ik1 , j 0 ∈ X ∩ Ik2 then
ωX (p) ≤ Ex∈Zl 1W ·(a(k1 )
(k )
(k )
ψ 1
(x)+h 1
N
j−Mk −1
1
j−Mk −1
1
j−Mk −1 )+b≡0
1
(mod p)
·1W ·(a(k2 )
(k2 )
(k2 )
j−Mk −1 ψj−Mk −1 (x)+hj−Mk −1 )+b≡0
2
2
2
P
(k1 )
(k )
For i = 1, 2, write aj−M
ψ 1
(x) = ts=1 Lki ,s xs
k1 −1 j−Mk1 −1
and our condition becomes
t
X
(k )
Lk1 ,s xs = −W −1 b − W −1 hj1 1 (mod p)
s=1
t
X
(k )
Lk2 ,s xs = −W −1 b − W −1 hj2 2
(mod p)
s=1
Now by assumptions, we have (Lk1 ,s )1≤s≤t and (Lk2 ,s )1≤s≤t are linearly independent. Now suppose Lj,t1 = λLj,t2 (mod p), Lj 0 ,t1 = λLj 0 ,t2 (mod p) for some λ ∈ Z then
Ljt1 Lj 0 t 1
≡ 0 (mod p)
|Ljt1 Lj 0 t2 − Ljt2 Lj 0 t1 | = Ljt2 Lj 0 t2 (mod p)
24
ÁKOS MAGYAR AND TATCHAI TITICHETRAKUN
L
Lj 0 t1 so p - jt1
0
Ljt2
Lj t2
L
Hence if N > Nk is sufficiently large then p > ω(N ) > jt1
Ljt2
This implies ωX (p) ≤ p−2
Lj 0 t1 .
Lj 0 t2 Remark A.3. If we choose W according to our last argument, Since the number of linear forms we’ll
consider is finite so we can choose W to be a fixed finite constant, independent of N .
The rest of the argument is very similar as in [8].Now after expanding Ep (z, z 0 ), we have
X (−1)|X|+|X 0 | ωX∪X 0 (p)
P
P
Ep (z, z 0 ) =
0
j∈X zj + j∈X 0 zj
p
0
X,X ⊆[M ]
M
X
0
0
= 1 − 1p>ω(n)
(p−1−zj + p−1−zj − p−1−zj −zj )
j=1
+
k
X
OM (p−2 )
X
0
1p>ω(N );p|∆i λ(i)
p (z, z ) +
X∪X 0 6⊆Iα
|X∪X 0 |>1
i=0
where
0
λ(i)
p (z, z ) =
X∪X 0 ⊆Ii
|X∪X 0 |>1
Ep(0) (z, z 0 ) = 1 +
j∈X
P
zj + j∈X 0 zj0
OM (p−1 )
X
Define
p
P
k
X
p
P
j∈X
zj +
P
j∈X 0
zj0
0
1p>ω(n);p|∆i λ(i)
p (z, z )
i=0
then write Ep =
Ep(1) (z, z 0 ) :=
Ep(2) (z, z 0 )
:=
(0) (1) (2) (3)
Ep Ep Ep Ep ,
where
Ep (z, z 0 )
(0)
Ep (z, z 0 )
M
Y
QM
j=1 (1
0
0
− 1p>ω(N ) p−1−zj )(1 − 1p>ω(N ) p−1−zj )(1 − 1p>ω(N ) p−1−zj −zj )−1
0
0
(1 − 1p≤ω(N ) p−1−zj )−1 (1 − 1p≤ω(N ) p−1−zj )−1 (1 − 1p≤ω(N ) p−1−zj −zj )
j=1
Ep(3) (z, z 0 )
:=
M
Y
0
0
(1 − p−1−zj )(1 − p−1−zj )(1 − p−1−zj −zj )−1
j=1
Let Gi =
Q
(i)
p Ep ,
noting that G3 =
ζ(1+zj +zj0 )
j=1 ζ(1+zj )ζ(1+zj0 ) .
QM
For σ > 0, define
DσM = {(zj , zj0 ) : <(zj ), <(zj0 ) ∈ (−σ, 100), 1 ≤ j ≤ M }
Now suppose f is analytic on DσM , define the norm
αM 0
α0M ∂ α1
∂
∂ α1
∂
kf kC k (DσM ) :=
sup
·
·
·
f
···
0
0
∂z1
∂zM
∂z1
∂zM
(α1 ,..,αM )
M)
L∞ (Dσ
0
0
P(α1 ,..,α
P M)
αi + α0i ≤k
1
Lemma A.3. [8] Let 0 < σ < 6M
then for i = 0, 1, 2, Gi is absolutely convergent in DσM and hence
represent an analytic function on this domain and we have
r Y
log R
kG0 kC r (DσM ) = OM
(1 + OM (p2M σ−1 )), 0 ≤ r ≤ M
log log R
Qk
p|
i=0
∆i
CORNERS IN DENSE SUBSETS OF Pd
25
1
kG0 kC M (DσM ) ≤ exp(OM (log 3 R))
kG1 kC M (DσM ) = OM (1)
kG2 kC M (DσM ) ≤ OM,W (1)
k Y
Y
G0 (0, 0) =
1
(1 + OM (p− 2 ))
i=0 p|∆i
G1 (0, 0) = 1 + OM (1)
M
W
G2 (0, 0) =
φ(W )
For the proof of this lemma see lemma 10.3 and lemma10.6 in [8] with ∆ =
G0 (0, 0) =
Y
Ep(0)
Qk
i=0 ∆i ;
noting that
k
k Y
Y
X
Y
(i)
=
(1 +
λp (0, 0)) ≤
(1 + |λp(i) (0, 0)|)
p|∆
i=0
p|∆
i=0 p|∆i
1
(i)
and we crudely have |λp (0, 0)| = 1 + OM (p− 2 ) Now the contour integral takes the form
0
Z
Z
M
Y
ζ(1 + zj + zj0 )Rzj +zj
−M
0
dzj dzj0
(2πi)
···
G(z, z )
0 )z 2 z 0 2
ζ(1
+
z
)ζ(1
+
z
j
Γ1
Γ1
j j j
j=1
with G = G0 G1 G2 . Now apply the following lemma (Lemma 10.4 in [8],see also [12],[13]) to prove the
estimate A1.
Lemma A.4 (Goldston-Yildirim [8],[12],[13]). Let R > 0, G(z, z 0 ) is analytic in DσM for some δ > 0 and
1
kG0 kC k (DσM ) = exp(OM,σ (log 3 R)) then
0
(2πi)
−M
Z
Z
···
Γ1
0
G(z, z )
Γ1
= G(0, ..., 0) logM R +
M
X
M
Y
ζ(1 + zj + zj0 )Rzj +zj
0
ζ(1 + zj )ζ(1 + zj0 )zj2 zj2
j=1
dzj dzj0
√
OM,σ (kG0 kC j (DσM ) ) logM −j R + OM,σ (exp(−δ R))
j=1
for some δ > 0.
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E-mail address: tatchai@math.ubc.ca