Geometric analog of Green-Tao’s PAP theorem Chunlei Liu Shanghai Jiao Tong University Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 1 / 22
Green-Tao’s prime arithmetic progression theorem Theorem (Green-Tao’s PAP theorem) The primes contain arbitrarily long arithmetic progressions. Definition An arithmetic progression is called a truncated residue class in Z . Theorem The primes contain arbitrarily large truncated residue classes. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 2 / 22
Truncated residue classes in polynomial rings Definition Let a, g ∈ F q [ t ] with g � = 0 , and r > 0 . Then { a + bg | b ∈ F q [ t ] , deg b < r } is called a truncated residue class in F q [ t ] . Theorem (Thai Hoang Le) The irreducible polynomials contain arbitrarily large truncated residue classes. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 3 / 22
Congruent functions C is a projective over F q . K C is the function field of C . D is a nonzero finite effective divisor on C . Definition Let f, g ∈ K C be functions which are regular at Supp( D ) . We call f ≡ g (mod D ) if ord P ( f − g ) ≥ ord P ( D ) , ∀ P ∈ Supp( D ) . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 4 / 22
Equivalent finite effective divisors D 0 , D 1 are nonzero finite effective divisors on C . Supp( D i ) ∩ Supp( D ) = ∅ , i = 0 , 1 . Definition We call D 0 ≡ D 1 (mod D ) if there is a nonzero function f ≡ 1(mod D ) such that D 0 − D 1 = ( f ) , where � ( f ) := ord P ( f ) · P. P ∤ ∞ Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 5 / 22
Truncated equivalence classes of finite effective divisors r is a positive integer. ord ∞ ( f ) = min P |∞ { ord P ( f ) } , f ∈ K C . Definition We call { ( f ) + D 0 | f ≡ 1(mod D ) , ord ∞ ( f ) < r } a truncated equivalence class of finite effective divisors modulo D . Theorem The prime divisors in every equivalence class contain arbitrarily large truncated equivalence classes of finite effective divisors. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 6 / 22
Congruent algebraic numbers K is a number field such as Q and Q ( i ) . O K is the ring of integers in K such as Z and Z [ i ] . m is a nonzero ideal of O K . Definition Let a, b ∈ K be numbers whose denominators are prime to m . we call a ≡ b (mod m ) if ord ℘ ( a − b ) ≥ ord ℘ ( m ) , ∀ ℘ | m . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 7 / 22
Equivalent classes of ideals a , b are nonzero ideals of O K prime to m . Definition We call a ≡ b (mod m ) if there is a nonzero number ξ ≡ 1(mod m ) such that a = ( ξ ) b . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 8 / 22
Truncated equivalence classes of ideals Min = � � x � 2 σ |∞ [ K σ : R ] � x � 2 σ , f ∈ K C . Definition We call { ( ξ ) a | ξ ≡ 1(mod m ) , � ξ � ∞ < r } a truncated equivalence class of ideals modulo D . Theorem The prime ideals in every equivalence class contain arbitrarily large truncated equivalence classes of ideals. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 9 / 22
Szemer´ edi’s theorem Theorem (Szemer´ edi, 1975) Any subset of the integers of upper positive density contains arbitrarily long arithmetic progressions. Theorem (Furstenberg-Katznelson, 1978) Any subset of the polynomials of upper positive density contains arbitrarily large truncated residue classes. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 10 / 22
Hyper-graphs from polynomials k is any fixed positive integer. B k is the set of polynomial of degree < k . e j = B k \ { j } . ( B k , { e j } j ∈ B k ) is a hyper-graph. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 11 / 22
Inverse systems on hyper-graphs I = { x ∈ F q [ t ] | ( x, y ) = 1 , ∀ y ∈ B k \ { 0 }} . { ( F q [ t ] / ( N )) e j } N ∈ I is an inverse system associated to e j . { ( F q [ t ] / ( N )) e j } N ∈ I,j ∈ B k is an inverse system associated to ( B k , { e j } j ∈ B k ) . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 12 / 22
Pseudo-random measures on inverse systems ν N,j ( j ∈ B k , N ∈ I ) is a nonnegative function on ( F q [ t ] / ( N )) e j . ˜ ν N ( N ∈ I ) is a nonnegative function on ( F q [ t ] / ( N )) . ˜ � ν N,j ( x ) = ˜ ˜ ν N ( ( i − j ) x i ) . i ∈ e j Theorem If { ˜ ν N } is k -pseudo-random, then { ˜ ν N,j } is pseudo-random. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 13 / 22
Pseudo-random measures on the polynomial ring ν r is a nonnegative function on the set of polynomials. ˜ ν N ( x ) = ν deg N ( x ) if deg x < deg N. Theorem If { ν r } is k -pseudo-random, then { ˜ ν N } is k -pseudo-random. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 14 / 22
Geometric relative Szemer´ edi’s theorem A r is a set of monic polynomials of degree r such that � 1 ν r ( g ) ≫ 1 , r → ∞ . q r − 1 g ∈ A r Theorem (Tao, 2005) If { ν r } is k -pseudo-random, then there exists a set A r that contains a truncated residue class of size q k . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 15 / 22
The truncated von Mangolt function � ( − 1) s , g is a product of s distinct irr. poly, µ F q ( t ) ( g ) = 0 , n otherwise. ϕ is a nonnegative smooth function on R . ϕ (0) = 1 and ϕ ( x ) = 0 if | x | > 1 . cr < R < rq − q 2 k . Definition The truncated von Mangolt function Λ F q ( t ) ,R of F q ( t ) is defined by the formula � µ F q ( t ) ( d ) ϕ (deg d Λ F q ( t ) ,R ( g ) := ) . R d | g,d monic Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 16 / 22
The normalized truncated von Mangolt function on a reduced residue class modulo W c log log r < w < log log r . W is the product of irreducible polynomials of degree < w . α is a polynomial prime to W with degree < deg W . � Λ F q ( t ) ,R,W,α ( g ) = c W,R,ϕ Λ F q ( t ) ,R ( Wg + α ) . Theorem The function � Λ 2 F q ( t ) ,R,W,α is k -pseudo-random. In particular, � 1 Λ 2 � F q ( t ) ,R,W,α ( g ) = 1 + o (1) , r → ∞ . q r deg g<r Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 17 / 22
Density of irreducible polynomials relative to � Λ 2 F q ( t ) ,W,α,R A r is the set of monic polynomials of degree < r such that Wg + α is irreducible. Theorem � 1 Λ 2 � F q ( t ) ,W,α,R ( g ) ≫ 1 , r → ∞ . q r g ∈ A r Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 18 / 22
The cross-correlation condition s ≤ q q k +1 , m ≤ q k +1 are arbitrary positive integers. ψ 1 , · · · , ψ s are arbitrary mutually independent linear forms in m variables whose coefficients are polynomials of degree < k . Definition The system { ν r } is said to satisfy the k -cross-correlation condition if s � � 1 ν r ( ψ j ( g ) + b j ) = 1 + o (1) , r → ∞ q m ( r − k ) j =1 deg gi<r − k i =1 , ··· ,m uniformly for all polynomials b 1 , · · · , b s of degree < r . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 19 / 22
The auto-correlation condition a is an arbitrary polynomial of degree < k . K is an arbitrary positive integer. Definition The system { ν r } is said to satisfy the k -auto-correlation condition if � 1 auto( ψ ( g ); ν r ◦ a ) K = O (1) , q r − k deg gi< deg r − k i =1 , ··· ,m where s � � 1 ν r ( ag + b i ) , b ∈ F q [ t ] s . auto( b ; ν r ◦ a ) = q r − k i =1 deg g<r − k Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 20 / 22
Pseudo-randomness Definition The system { ν r } is k -pseudo-random if it satisfies the k -cross-correlation condition and the k -auto-correlation condition. Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 21 / 22
Relative multi-dimensional Szemer´ edi’s theorem Theorem (Relative multidimensional Szemer´ edi theorem) Let Z and Z ′ be two finite additive groups depending on a positive integer parameter N tending to infinity, ν a measure on Z ′ , and φ = ( φ j ) j ∈ J a finite collection of group homomorphisms from Z to Z ′ . Suppose that φ is ergodic, and ν φ is a pseudorandom family of measures. If A is a subset of Z ′ such that � � 1 1 A ( x + φ j ( r )) ν ( x + φ j ( r )) ≤ δ | Z | · | Z ′ | x ∈ Z ′ ,r ∈ Z j ∈ J for some 0 < δ ≤ 1 , then | A | | Z ′ | = o δ → 0; | J | (1) + o N →∞ ; δ (1) . Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 22 / 22
Recommend
More recommend
