Partition regularity of some quadratic equations
Joint work with N. Frantzikinakis
Ergodic Theory with Connections to Arithmetic
Heraklion, June 3, 2013
– I –
A Theorem of partition regularity
Definition.
A family of finite subsets of N is partition regular if, for every finite partition
N = C1 ∪ · · · ∪ Cr
of N, at least one element Cj of this partition contains a set in this family.
The notion of partition regularity of an equation f (x1, . . . , xk ) = 0 with integer
unknowns is defined in a similar way.
We restrict to non trivial solutions, that is with distinct values of x1, . . . , xk .
1
Examples.
– Schur Theorem (1916): The equation x + y = z is partition regular.
– van der Waerden Theorem (1927): The family of arithmetic progressions of
given length is partition regular.
– Rado’s Theorem (1933) characterizes the systems of linear equations that
are partition regular.
– Less is known for non linear equations.
Classical problem: are the equations x2 + y 2 = z 2 and x2 + y 2 = 2z 2 partition
regular?
– The polynomial van der Waerden Theorem of Bergelson and Leibman (1996)
provides examples of non linear configurations. These configurations are invariant under translations.
2
Definition (Partition regularity of an equation with a free variable).
The equation p(x, y, λ) = 0 is partition regular if, for every finite partition
N = C1 ∪ · · · ∪ Cr
of N there exist x 6= y in the same subset Cj and an arbitrary λ ∈ N with
p(x, y, λ) = 0.
Examples.
– The equation x − y = λ2 is partition regular (Sárközy 1978, Furstenberg).
This is a particular case of the polynomial van der Waerden Theorem.
– The equation x + y = λ2 is partition regular (Khalfalah & Szemerédi 2006).
In this talk we consider equations of the form p(x, y, λ) = 0 where p is a quadratic
homogeneous polynomial.
These equations are not translation invariant and not linear in x and y.
3
Open questions:
Are the equations x2 + y 2 = λ2 and x2 + y 2 = 2λ2 partition regular?
Theorem
The equations:
16x2 + 9y 2 = λ2
and
x2 + y 2 − xy = λ2
(and many others) are partition regular.
In the sequel we restrict to the equation 16x2 + 9y 2 = λ2.
4
– II –
Reduction to a theorem
about multiplicative functions
Parametrization
We obtain a parametrization of the solutions of the equation 16x2 + 9y 2 = λ2
by letting
x = km(m + 3n)
;
y = k(m + n)(m − 3n)
and λ = k(5m2 + 9n2 + 6mn) for acceptable k, m, n ∈ N, meaning such that x
and y are positive and distinct.
This parametrization satisfies:
• it is invariant under dilation;
• x and y are products of two linear forms in the variables m and n.
• m has the same coefficient in the four linear forms.
It is possible to explicitly characterize the homogeneous quadratic equations
p(x, y, λ) = 0 admitting a similar parametrization of the family of solutions.
The results of this talk are valid for all these equations.
5
Partition regularity and density
Translation invariant equations that are partition regular satisfy often the stronger
property of density regularity, meaning that the equation admits a solution in
every set of integers of positive density.
Furstenberg’s correspondence principle allows to deduce density regularity from
a recurrence result in ergodic theory.
The equation
6x2 + 9y 2 = λ2
is not translation invariant.
Open question.
Is it true that every set of integers of positive density contains a non trivial
solution (x, y)?
6
Multiplicative density
The equation 6x2 + 9y 2 = λ2 is invariant under dilations, and this leads us to
use the multiplicative density.
Let {p1 < p2 < p3 < . . . } be the set of primes. For every N , let
ΦN =
m
m m
p1 1 p2 2 . . . pN N : 0 ≤ m1, m2, . . . , mN < N
n
o
.
(ΦN : N ≥ 1) is an example of a multiplicative Følner sequence: for every r ∈ Q+,
ΦN \ rΦN |ΦN |
→ 0 when N → +∞
where rΦN = {rx : x ∈ ΦN } ∩ N.
Definition.
The multiplicative density of the subset E of N is
|E ∩ ΦN |
dmult(E) = lim sup
.
|ΦN |
N →+∞
7
Since the multiplicative density is subadditive, partition regularity follows from:
Theorem (Multiplicative density regularity).
Every subset E of N with positive multiplicative density contains a non trivial
solution (x, y) of the equation 16x2 + 9y 2 = λ2.
8
A translation to ergodic theory
Definition.
A measure preserving action of the multiplicative group Q+ on a probability
space (X, µ) is a family (Tr : r ∈ Q+), of measurable, invertible, measure preserving transformations of X with
Tr Tr = Trs for all r, s ∈ Q+ .
Multiplicative version of Furstenberg’s correspondence principle.
Let E ⊂ N be a set of positive multiplicative density.
There exist a measure preserving action (Tr : r ∈ Q+) of Q+ on a probability
space (X, µ) and a subset A of X with
µ(A) = dmult(E)
and, for every k ∈ N and all r1, . . . , rk ∈ Q+,
dmult(r1E ∩ · · · ∩ rk E) ≥ µ(Tr1 A ∩ · · · ∩ Trk A) .
9
We are reduced to show:
Theorem (Ergodic formulation).
Let (Tr : r ∈ Q+) be a measure preserving action of Q+ on a probability space
(X, µ) and let A ⊂ X be a set of positive measure.
Then there exist x 6= y ∈ N and λ ∈ N with 16x2 + 9y 2 = λ2 such that
µ(Tx−1A ∩ Ty−1A) > 0 .
Indeed, if (X, µ), (Tr : r ∈ Q+) and A are given by the correspondence principle,
for x, y ∈ N we have
dmult({k ∈ N : kx ∈ E and ky ∈ E}) = dmult(x−1E ∩ y −1E) ≥ µ(Tx−1A ∩ Ty−1A).
10
Let (X, µ), Tr and A be as in the last Theorem. We want to show that there
exists a non trivial solution (x, y) of the equation such that
−1
−1
µ Tx A ∩ Ty A > 0.
Using the parametrization and the invariance of µ under the transformations Tr ,
we are reduced to showing that there exist acceptable m, n ∈ N with
Z
X
Tm(m+3n)(m+n)−1(m−3n)−1 f · f dµ > 0
where f = 1A.
We recognize an integral arising in the Spectral Theorem.
11
Multiplicative functions
Definition.
A multiplicative function is a function χ : N → C, of modulus 1, such that
χ(xy) = χ(x)χ(y) for all x, y ∈ N .
We write M for the family of multiplicative functions.
These functions are often called “completely multiplicative functions”.
A multiplicative function is characterized by its value on the primes.
Endowed with the pointwise multiplication and with the topology of the pointwise convergence, the family M is a compact abelian group.
Its unit is the constant function 1.
12
Every multiplicative function can be extended to a function on Q+ by
χ a/b) = χ(a) χ(b) for all a, b ∈ N.
The group M of multiplicative functions is the dual group of the multiplicative
group Q+, the duality being given by the last formula.
Spectral Theorem of actions of Q+.
Let (Tr : r ∈ Q+) be a measure preserving action of Q+ on a probability space
(X, µ) and let f ∈ L2(µ).
Then there exists a finite positive measure ν on the compact abelian group M,
called the spectral measure of f , such that
Z
X
Tr f · f dµ =
Z
M
χ(r) dν(χ) for every r ∈ Q+.
13
Let A ⊂ X with µ(A) > 0. Let ν be the spectral measure of f = 1A.
Recall that we want to show that there exist acceptable m, n ∈ N such that
Z
X
Tm(m+3n)(m+n)−1(m−3n)−1 f · f dµ > 0.
By the Spectral Theorem and the multiplicativity of the functions χ, we are
reduced to show:
Theorem (Spectral formulation).
Let (Tr : r ∈ Q+) be a measure preserving action of Q+ on a probability space
(X, µ), A ⊂ X a set of positive measure and ν the spectral measure of 1A.
Then there exist m, n ∈ N with m > 3n, m(m + 3n) 6= (m + n)(m − 3n) and
Z
M
χ(m)χ(m + 3n)χ(m + n)χ(m − 3n) dν(χ) > 0.
(This integral is allways ≥ 0.)
14
Averaging
In fact we show that there are many values of m and n such that this integral
is positive: we show that some average of this integral is positive:
Since we are proving a result about sets of positive multiplicative density, multiplicative averages may seem more natural, but we use ordinary (additive)
averages.
Some notation
If φ is a function defined on a finite set A,
1 X
φ(x).
Ex∈A φ(x) =
|A| x∈A
Same notation for a function of several variables.
For every N ∈ N,
[N ] = {1, 2, . . . , N }
;
ZN = Z/N Z.
15
We find it more convenient to deal with functions defined on a cyclic group
than on an interval of N.
Notation.
f for the smallest prime ≥ 10N .
For every N ∈ N, we write N
For χ ∈ M and x ∈ ZN
e,
χN (x) =

χ(x)
0
if x ∈ [N ] ;
otherwise.
f = N and
This is only a technical point, you can forget it and consider that N
that χN = χ.
16
Taking averages in the preceding formulation, we get that the theorem of partition regularity follows from:
Theorem (Final form).
Let (Tr : r ∈ Q+) be a measure preserving action of Q+ on a probability space
(X, µ), A ⊂ X a set of positive measure and ν the spectral measure of 1A.
Then the lim sup when N → +∞ of
Z
M
Em,n∈Z 1[N ](n) χN (m) χN (m + 3n) χN (m + n) χN (m − 3n) dν(χ)
e
N
is positive.
17
– III –
Fourier analysis of multiplicative functions
The first tool for the analysis of the expression in the Theorem is the discrete
Fourier transform.
Definition
The discrete Fourier coefficients of a function φ on ZN are
fb(ξ) = Ex∈ZN φ(x) e(−xξ/N )
for ξ ∈ ZN
where e(x) = exp(2πix).
Contrasting with the Möbius function, multiplicative functions can have large
Fourier coefficients and even large averages. However, their Fourier coefficients
are small on “minor arcs”.
18
A Lemma of Kátai (1986) gives:
Corollary (Kátai).
For every θ > 0 there exist q ∈ N and c > 0, depending only on θ, such that, for
every χ ∈ M,
p
c
if sup En∈[N ] χ(n) e(nα) > θ then there exists p with α − <
.
q
N
χ∈M
This could be the starting point of the circle method.
For each χ ∈ M, χN can be decomposed as a sum of
• a structured term, meaning approximatively periodic with small period,
• plus a Fourier uniform term, meaning with small Fourier coefficients.
19
Fourier uniformity is measured by the Gowers U 2-norm kφkU 2 of φ.
Definition
The U 2-norm of a function φ on ZN is
kφkU 2 = Ex,t1,t2∈ZN φ(x) φ(x + t1) φ(x + t2) φ(x + t1 + t2)
=
X
1/4
1/4
4
b
|φ(ξ)|
.
ξ∈ZN
If |f | ≤ 1,
b(ξ)| ≤ kf k .
kf k2
≤
max
|
f
2
U2
U
20
– IV –
“Higher order Fourier analysis” of multiplicative functions
This Fourier analysis of multiplicative functions is not sufficient.
Indeed, we want to study
Em,n∈Z 1[N ](n) χN (m) χN (m + 3n) χN (m + n) χN (m − 3n).
e
N
But expressions of this form are not controlled by the U 2-norms of the functions
in the average.
We need the Gowers U 3-norm, that we do not define here.
Lemma.
If φ1, . . . , φ4 are functions on ZN of modulus ≤ 1 then
1/2
1[N ](n)φ1(m) φ2(m + 3n) φ3(m + n) φ4(m − 3n) ≤ C min kφikU 3 .
e
N
i
Em,n∈Z
21
A theorem of decomposition
Our main result is a theorem of decomposition of multiplicative functions for
the U 3-norm.
Weak decomposition theorem for the U 3-norm (simplified statement)
For every > 0 there exist q ∈ N and c > 0, independent of χ and of N , such
that, for every χ ∈ M, there is a decomposition of χN as a sum
χN (x) = χs(x) + χu(x) for x ∈ ZN
e
where
c
for every x ∈ ZN
χs(x + q) − χs(x) ≤
(Structured part)
kχukU 3 ≤ .
(U 3-uniform part)
N
and
22
Some ingredients in the proof of the decomposition theorem:
• The Inverse Theorem for the U 3-norm of Green & Tao (2005) leads to study
the correlation of multiplicative functions with some nilsequences.
• The number theoretic input is again Kátai’s Lemma, now applied to nilsequences.
• At this point, we need another “dynamical” ingredient, the Theorem of quantitative equidistribution of orbits in nilmanifolds of Green & Tao (2007).
23
– V –
Putting the pieces together
We want to show that the lim sup of
Z
Em,n∈Z 1[N ](n)χN (m)χN (m + 3n)χN (m + n)χN (m − 3n) dν(χ)
e
N
M
is positive.
Strategy.
• We need a more elaborate form of the decomposition theorem, with a stronger
control of the uniformity norm and an “explicit” form of the structured part.
• The price to pay is the introduction of a third “error” term.
• The uniform term has a negligible contribution.
• We need also to use the fact that the measure ν on M is not an arbitrary
one: it is the spectral measure of some nonnegative function on X.
• This allows us to change the set on which the averages are taken, and to
eliminate the error term.
24
Thank you for your attention
Regular partition of a bottle of ouzo