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 = C 1 ∪ · · · ∪ C r of N , at least one element C j of this partition contains a set in this family. The notion of partition regularity of an equation f ( x 1 , . . . , x k ) = 0 with integer unknowns is defined in a similar way. We restrict to non trivial solutions, that is with distinct values of x 1 , . . . , x k . 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 x 2 + y 2 = z 2 and x 2 + y 2 = 2 z 2 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. 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 = C 1 ∪ · · · ∪ C r of N there exist x � = y in the same subset C j and an arbitrary λ ∈ N with p ( x, y, λ ) = 0. Examples. – The equation x − y = λ 2 is partition regular (S´ ark¨ ozy 1978, Furstenberg). This is a particular case of the polynomial van der Waerden Theorem. – The equation x + y = λ 2 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 . 3
Open questions: Are the equations x 2 + y 2 = λ 2 and x 2 + y 2 = 2 λ 2 partition regular? Theorem The equations: 16 x 2 + 9 y 2 = λ 2 x 2 + y 2 − xy = λ 2 and (and many others) are partition regular. In the sequel we restrict to the equation 16 x 2 + 9 y 2 = λ 2 . 4
– II – Reduction to a theorem about multiplicative functions
Parametrization We obtain a parametrization of the solutions of the equation 16 x 2 + 9 y 2 = λ 2 by letting x = km ( m + 3 n ) ; y = k ( m + n )( m − 3 n ) and λ = k (5 m 2 + 9 n 2 + 6 mn ) 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 6 x 2 + 9 y 2 = λ 2 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 6 x 2 + 9 y 2 = λ 2 is invariant under dilations, and this leads us to use the multiplicative density. Let { p 1 < p 2 < p 3 < . . . } be the set of primes. For every N , let � � p m 1 1 p m 2 . . . p m N Φ N = : 0 ≤ m 1 , m 2 , . . . , m N < N . 2 N (Φ N : N ≥ 1) is an example of a multiplicative Følner sequence: for every r ∈ Q + , � � � � � Φ N \ r Φ N � → 0 when N → + ∞ | Φ N | where r Φ N = { rx : x ∈ Φ N } ∩ N . Definition. The multiplicative density of the subset E of N is | E ∩ Φ N | d mult ( E ) = lim sup . | Φ N | 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 16 x 2 + 9 y 2 = λ 2 . 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 ( T r : r ∈ Q + ), of measurable, invertible, measure pre- serving transformations of X with T r T r = T rs 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 ( T r : r ∈ Q + ) of Q + on a probability space ( X, µ ) and a subset A of X with µ ( A ) = d mult ( E ) and, for every k ∈ N and all r 1 , . . . , r k ∈ Q + , d mult ( r 1 E ∩ · · · ∩ r k E ) ≥ µ ( T r 1 A ∩ · · · ∩ T r k A ) . 9
We are reduced to show: Theorem (Ergodic formulation). Let ( T r : 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 16 x 2 + 9 y 2 = λ 2 such that µ ( T − 1 A ∩ T − 1 A ) > 0 . x y Indeed, if ( X, µ ), ( T r : r ∈ Q + ) and A are given by the correspondence principle, for x, y ∈ N we have d mult ( { k ∈ N : kx ∈ E and ky ∈ E } ) = d mult ( x − 1 E ∩ y − 1 E ) ≥ µ ( T − 1 A ∩ T − 1 A ) . x y 10
Let ( X, µ ), T r 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 � � T − 1 A ∩ T − 1 µ A > 0 . x y Using the parametrization and the invariance of µ under the transformations T r , we are reduced to showing that there exist acceptable m, n ∈ N with � X T m ( m +3 n )( m + n ) − 1 ( m − 3 n ) − 1 f · f dµ > 0 where f = 1 A . 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 point- wise 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 ( T r : r ∈ Q + ) be a measure preserving action of Q + on a probability space ( X, µ ) and let f ∈ L 2 ( µ ). Then there exists a finite positive measure ν on the compact abelian group M , called the spectral measure of f , such that � � X T r f · f dµ = 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 = 1 A . Recall that we want to show that there exist acceptable m, n ∈ N such that � X T m ( m +3 n )( m + n ) − 1 ( m − 3 n ) − 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 ( T r : 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 1 A . Then there exist m, n ∈ N with m > 3 n , m ( m + 3 n ) � = ( m + n )( m − 3 n ) and � M χ ( m ) χ ( m + 3 n ) χ ( m + n ) χ ( m − 3 n ) 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, 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 , � E x ∈ A φ ( x ) = 1 φ ( x ) . | A | x ∈ A Same notation for a function of several variables. For every N ∈ N , [ N ] = { 1 , 2 , . . . , N } ; Z N = 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. For every N ∈ N , we write � N for the smallest prime ≥ 10 N . For χ ∈ M and x ∈ Z � N , χ ( x ) if x ∈ [ N ] ; χ N ( x ) = 0 otherwise. This is only a technical point, you can forget it and consider that � N = N and that χ N = χ . 16
Taking averages in the preceding formulation, we get that the theorem of par- tition regularity follows from: Theorem (Final form). Let ( T r : 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 1 A . Then the lim sup when N → + ∞ of � N 1 [ N ] ( n ) χ N ( m ) χ N ( m + 3 n ) χ N ( m + n ) χ N ( m − 3 n ) dν ( χ ) M E m,n ∈ Z � is positive. 17
– III – Fourier analysis of multiplicative functions
Recommend
More recommend
