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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 ∈ Fq[t] with g = 0, and r > 0. Then {a + bg | b ∈ Fq[t], deg b < r} is called a truncated residue class in Fq[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 Fq. KC is the function field of C. D is a nonzero finite effective divisor on C.

Definition

Let f, g ∈ KC be functions which are regular at Supp(D). We call f ≡ g(modD) if

• rdP (f − g) ≥ ordP (D), ∀P ∈ Supp(D).

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 4 / 22

Equivalent finite effective divisors

D0, D1 are nonzero finite effective divisors on C. Supp(Di) ∩ Supp(D) = ∅, i = 0, 1.

Definition

We call D0 ≡ D1(modD) if there is a nonzero function f ≡ 1(modD) such that D0 − D1 = (f), where (f) :=

• P∤∞
• rdP (f) · 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.

• rd∞(f) = minP|∞{ordP (f)}, f ∈ KC.

Definition

We call {(f) + D0 | f ≡ 1(modD), 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). OK is the ring of integers in K such as Z and Z[i]. m is a nonzero ideal of OK.

Definition

Let a, b ∈ K be numbers whose denominators are prime to m. we call a ≡ b(modm) if

• rd℘(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 OK prime to m.

Definition

We call a ≡ b(modm) if there is a nonzero number ξ ≡ 1(modm) 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

x2

Min = σ|∞[Kσ : R]x2 σ, f ∈ KC.

Definition

We call {(ξ)a | ξ ≡ 1(modm), 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. Bk is the set of polynomial of degree < k. ej = Bk \ {j}. (Bk, {ej}j∈Bk) 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 ∈ Fq[t] | (x, y) = 1, ∀y ∈ Bk \ {0}}. {(Fq[t]/(N))ej}N∈I is an inverse system associated to ej. {(Fq[t]/(N))ej}N∈I,j∈Bk is an inverse system associated to (Bk, {ej}j∈Bk).

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 12 / 22

Pseudo-random measures on inverse systems

˜ νN,j (j ∈ Bk, N ∈ I) is a nonnegative function on (Fq[t]/(N))ej. ˜ νN (N ∈ I) is a nonnegative function on (Fq[t]/(N)). ˜ νN,j(x) = ˜ νN(

• i∈ej

(i − j)xi).

Theorem

If {˜ νN} is k-pseudo-random, then {˜ νN,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

νr is a nonnegative function on the set of polynomials. ˜ νN(x) = νdeg N(x) if deg x < deg N.

Theorem

If {νr} is k-pseudo-random, then {˜ νN} 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

Ar is a set of monic polynomials of degree r such that 1 qr−1

• g∈Ar

νr(g) ≫ 1, r → ∞.

Theorem (Tao, 2005)

If {νr} is k-pseudo-random, then there exists a set Ar that contains a truncated residue class of size qk.

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 15 / 22

The truncated von Mangolt function

µFq(t)(g) = (−1)s, g is a product of s distinct irr. poly, 0, n otherwise. ϕ is a nonnegative smooth function on R. ϕ(0) = 1 and ϕ(x) = 0 if |x| > 1. cr < R < rq−q2k.

Definition

The truncated von Mangolt function ΛFq(t),R of Fq(t) is defined by the formula ΛFq(t),R(g) :=

• d|g,d monic

µFq(t)(d)ϕ(deg d R ).

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.

• ΛFq(t),R,W,α(g) = cW,R,ϕΛFq(t),R(Wg + α).

Theorem

The function Λ2

Fq(t),R,W,α is k-pseudo-random. In particular,

1 qr

• deg g<r
• Λ2

Fq(t),R,W,α(g) = 1 + o(1), r → ∞.

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 17 / 22

Density of irreducible polynomials relative to Λ2

Fq(t),W,α,R

Ar is the set of monic polynomials of degree < r such that Wg + α is irreducible.

Theorem

1 qr

• g∈Ar
• Λ2

Fq(t),W,α,R(g) ≫ 1, r → ∞.

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 18 / 22

The cross-correlation condition

s ≤ qqk+1, m ≤ qk+1 are arbitrary positive integers. ψ1, · · · , ψs are arbitrary mutually independent linear forms in m variables whose coefficients are polynomials of degree < k.

Definition

The system {νr} is said to satisfy the k-cross-correlation condition if 1 qm(r−k)

• deg gi<r−k

i=1,··· ,m s

• j=1

νr(ψj(g) + bj) = 1 + o(1), r → ∞ uniformly for all polynomials b1, · · · , bs 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 {νr} is said to satisfy the k-auto-correlation condition if 1 qr−k

• deg gi<deg r−k

i=1,··· ,m

auto(ψ(g); νr ◦ a)K = O(1), where auto(b; νr ◦ a) = 1 qr−k

• deg g<r−k

s

• i=1

νr(ag + bi), b ∈ Fq[t]s.

Chunlei Liu (Shanghai Jiao Tong University)Geometric analog of Green-Tao’s PAP theorem 20 / 22

Pseudo-randomness

Definition

The system {νr} 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 = (φj)j∈J a finite collection of group homomorphisms from Z to Z′. Suppose that φ is ergodic, and νφ is a pseudorandom family of measures. If A is a subset of Z′ such that 1 |Z| · |Z′|

• x∈Z′,r∈Z
• j∈J

1A(x + φj(r))ν(x + φj(r)) ≤ δ for some 0 < δ ≤ 1, then |A| |Z′| = oδ→0;|J|(1) + oN→∞;δ(1).

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