Partition regularity of some quadratic equations Joint work with N. - - PowerPoint PPT Presentation

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

• f 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.

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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 + y2 = z2 and x2 + y2 = 2z2 partition regular? – The polynomial van der Waerden Theorem of Bergelson and Leibman (1996) provides examples of non linear configurations. These configurations are invari- ant under translations.

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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

• f N there exist x = 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 (S´ ark¨

• zy 1978, Furstenberg).

This is a particular case of the polynomial van der Waerden Theorem. – The equation x + y = λ2 is partition regular (Khalfalah & Szemer´ edi 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.

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Open questions: Are the equations x2 + y2 = λ2 and x2 + y2 = 2λ2 partition regular? Theorem The equations: 16x2 + 9y2 = λ2 and x2 + y2 − xy = λ2 (and many others) are partition regular. In the sequel we restrict to the equation 16x2 + 9y2 = λ2.

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– II – Reduction to a theorem about multiplicative functions

Parametrization We obtain a parametrization of the solutions of the equation 16x2 + 9y2 = 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.

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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 + 9y2 = 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)?

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Multiplicative density The equation 6x2 + 9y2 = 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 =

• pm1

1 pm2 2

. . . pmN

N

: 0 ≤ m1, m2, . . . , mN < N

• .

(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 dmult(E) = lim sup

N→+∞

|E ∩ ΦN| |ΦN| .

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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 + 9y2 = λ2.

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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 pre- serving transformations of X with TrTr = 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 ∩ · · · ∩ rkE) ≥ µ(Tr1A ∩ · · · ∩ TrkA) .

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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 = y ∈ N and λ ∈ N with 16x2 + 9y2 = λ2 such that µ(T −1

x

A ∩ T −1

y

A) > 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(T −1

x

A ∩ T −1

y

A).

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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 µ

• T −1

x

A ∩ T −1

y

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

• X Tm(m+3n)(m+n)−1(m−3n)−1f · f dµ > 0

where f = 1A. We recognize an integral arising in the Spectral Theorem.

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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 point- wise convergence, the family M is a compact abelian group. Its unit is the constant function 1.

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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

• X Trf · f dµ =
• M χ(r) dν(χ) for every r ∈ Q+.

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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

• X Tm(m+3n)(m+n)−1(m−3n)−1f · 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) = (m + n)(m − 3n) and

• M χ(m)χ(m + 3n)χ(m + n)χ(m − 3n) dν(χ) > 0.

(This integral is allways ≥ 0.)

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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, mul- tiplicative averages may seem more natural, but we use ordinary (additive) averages. Some notation If φ is a function defined on a finite set A, Ex∈A φ(x) = 1 |A|

• x∈A

φ(x). Same notation for a function of several variables. For every N ∈ N, [N] = {1, 2, . . . , N} ; ZN = Z/NZ.

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We find it more convenient to deal with functions defined on a cyclic group than on an interval of N. Notation. For every N ∈ N, we write N for the smallest prime ≥ 10N. For χ ∈ M and x ∈ Z

N,

χN(x) =

  

χ(x) if x ∈ [N] ;

• therwise.

This is only a technical point, you can forget it and consider that N = N and that χN = χ.

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Taking averages in the preceding formulation, we get that the theorem of par- tition 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

• M Em,n∈Z

N 1[N](n) χN(m) χN(m + 3n) χN(m + n) χN(m − 3n) dν(χ)

is positive.

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– 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

• f(ξ) = Ex∈ZN φ(x) e(−xξ/N)

for ξ ∈ ZN where e(x) = exp(2πix). Contrasting with the M¨

• bius function, multiplicative functions can have large

Fourier coefficients and even large averages. However, their Fourier coefficients are small on “minor arcs”.

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A Lemma of K´ atai (1986) gives: Corollary (K´ atai). For every θ > 0 there exist q ∈ N and c > 0, depending only on θ, such that, for every χ ∈ M, if sup

χ∈M

• En∈[N] χ(n) e(nα)
• > θ then there exists p with
• α − p

q

• < c

N . 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.

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Fourier uniformity is measured by the Gowers U2-norm φU2 of φ. Definition The U2-norm of a function φ on ZN is φU2 =

• Ex,t1,t2∈ZN φ(x) φ(x + t1) φ(x + t2) φ(x + t1 + t2)

1/4

=

ξ∈ZN

| φ(ξ)|41/4. If |f| ≤ 1, f2

U2 ≤ max |

f(ξ)| ≤ fU2.

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– 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

N 1[N](n) χN(m) χN(m + 3n) χN(m + n) χN(m − 3n).

But expressions of this form are not controlled by the U2-norms of the functions in the average. We need the Gowers U3-norm, that we do not define here. Lemma. If φ1, . . . , φ4 are functions on ZN of modulus ≤ 1 then

• Em,n∈Z

N 1[N](n)φ1(m) φ2(m + 3n) φ3(m + n) φ4(m − 3n)

• ≤ C min

i

φi1/2

U3 .

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A theorem of decomposition Our main result is a theorem of decomposition of multiplicative functions for the U3-norm. Weak decomposition theorem for the U3-norm (simplified statement) For every ǫ > 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 ∈ Z

N

where

• χs(x + q) − χs(x)
• ≤ c

N for every x ∈ ZN (Structured part) and χuU3 ≤ ǫ. (U3-uniform part)

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Some ingredients in the proof of the decomposition theorem:

• The Inverse Theorem for the U3-norm of Green & Tao (2005) leads to study

the correlation of multiplicative functions with some nilsequences.

• The number theoretic input is again K´

atai’s Lemma, now applied to nilse- quences.

• At this point, we need another “dynamical” ingredient, the Theorem of quan-

titative equidistribution of orbits in nilmanifolds of Green & Tao (2007).

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– V – Putting the pieces together

We want to show that the lim sup of

• M Em,n∈Z

N 1[N](n)χN(m)χN(m + 3n)χN(m + n)χN(m − 3n) dν(χ)

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
• ne: 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.

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Thank you for your attention

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