[PPT] - 2015 Difference set of primes and related problems Wen Huang PowerPoint Presentation

SLIDE 1

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

2015 Difference set of primes and related problems

Wen Huang Joint work with Xiaosheng Wu

Dept of Math., SCU+USTC

April, 2015

SLIDE 2

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Outlineµ

1

Background

2

A conjecture

3

Bohr problem

4

Main result

SLIDE 3

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Background

Given a subset S = {s1 < s2 < · · · } of natural numbers N = {1, 2, · · · }, we may consider it as a ”big set” as following:

1

S is called syndetic, if there is L ∈ N such that {i, i + 1, · · · , i + L − 1} ∩ S = ∅ for any i ∈ {1, 2, · · · };

2

S has positive upper Banach density, if BD∗(S) = lim sup

N→+∞

sup

M∈N

1 N |S ∩ {M, M + 1, · · · , M + N − 1}| is larger than zero.

3

S satisfies Erd¨

s condition, if ∞

i=1 1 si = +∞. For

example: Prime numbers P.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Backgroud

A basic qustion: µDoes a ”big subset” of natural numbers contain a ”good” linear structure? egµAre there any arbitrarily long arithmetic progressions in Prime numbers set P? In 1939 van der Corput prove the case k = 3 for prime set P. Szemer´ edi Theorem(1975§Cominatorial method): A subset of natural numbers with positive upper Banach density contains arithmetic progressions of arbitrary finite length.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Background

In fact, Szemer´ edi proved that for any δ > 0 and k ≥ 3, there is a minimal natural number N(δ, k) such that If N ≥ N(δ, k) and A ⊆ {1, 2, · · · , N} with |A| ≥ δN, then A contains an arithmetic progression of length k. (In 1953, Roth proved the case k=3.) In 1977, Furstenberg gave a ergodic theory’s proof for Szemer´ edi Theorem.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Background

Gowers(1998,2001) gave a new proof for Szemer´ edi Theorem, in particular he proved N(δ, k) ≤ 22δ−ck , ck = 22k+9. For k = 3, Roth Theorem(1953), Heath-Brown(1987), Szemer´ edi(1990). Bourgain(1999) proved: N(δ, 3) ≤ 2Cδ−2 log( 1

δ ).

Bourgain(2003), Green(2005) ect..

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Background

In 2008, Green and Tao proved (Szemer´ edi.s theorem in the primes). Let A be any subset of the prime numbers of positive relative upper density: if lim supN→+∞

|A∩{1,2,··· ,N}| |P∩{1,2,··· ,N}| > 0, then A contains infinitely

many arithmetic progressions of length k for all k. Green,Tao,Ziegler...... Erd¨

s conjectureµ Suppose that A satisfies Erd¨
s

condition, then A contains arbitrarily long arithmetic progressions.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
1. Background: our question

Now our question isµ Given k ≥ 1. If S is a /big set0of natural numbers, then all common differences n of arithmetic progressions a, a + n,a + 2n, · · · , a + kn with length k + 1 appeared in the subset S forms a set Ck(S) := {n ∈ N : S ∩ (S − n) ∩ (S − 2n) ∩ · · · ∩ (S − kn) = ∅}. How about the structure of the set Ck(S)? For exampleµFor prime numbers P, Is Ck(P) syndetic? Recall that Ck(P) is the set formed by all common differences of arithmetic progressions with length k + 1 appeared in P.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
2. A conjecture

In fact, we have the following cojectureµ Conjecture Given k ≥ 1. If S is a /big set0of natural numbers, then Ck(S) := {n ∈ N : S ∩ (S − n) ∩ (S − 2n) ∩ · · · ∩ (S − kn) = ∅} is an ”almost” Nilk Bohr0 set?

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
2. A conjecture

Where a subset A of natural numbers is called Nilk Bohr0 set, if there exists a k-step nilsystem (X, T), x0 ∈ X and an

pen neighborhood U of x0 such that

A ⊇ {n ∈ N : Tnx0 ∈ U}. Remark Nilk Bohr0 set is also called Bohr0 set; Bohr0 set is a classic notion; Nilk Bohr0 sets were introduced by Host and Kra recently; Each Nilk Bohr0 set is syndetic.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

k-nilsystem

Definition Let G be a k-step nilpotent Lie group and Γ a discrete cocompact subgroup of G. The compact manifold X = G/Γ is called a k-step nilmanifold. The group G acts on X by left translations and we write this action as (g, x) → gx. The Haar measure µ of X is the unique probability measure on X invariant under this action. Let τ ∈ G and T be the transformation x → τx of X, i.e the rotation induced by τ ∈ G. Then (X, T, µ) is called a k-step nilsystem.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

k-nilsystem

Remark Rotation on torus is 1-step nilsystem. Katznelson(2001) proved: A subset A of natural numbers is Bohr0 set if and

nly if ∃a1, a2, · · · , am ∈ R and ǫ > 0 such that

A ⊇

m

i=1

{n ∈ N : nai(mod Z) ∈ (−ǫ, ǫ)}. In later, one of Huang+Shao+Ye results is to give a similar characterization of Nilk Bohr0 set by generalized polynomials of degree k.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

3.Reasons of ConjectureµHigher order Bohr problem

In 1968, W.Veech provedµ

Theorem (Veech) Let S ⊆ N be a syndetic set. Then S − S := {s1 − s2 : s1, s2 ∈ S, s1 > s2} (That is, the common differences of arithmetic progressions with length 2 appeared in S) is an /almost0Bohr0 set, More precise, there exists a set M with BD∗(M) = 0 such that (S − S) ∪ M is a Bohr0 set.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Bohr Problem

Problem (Bohr problem) Let S ⊆ N be a syndetic set. Is S − S a Bohr0 set? Furstenberg(1981), Glasner(1998,2004), Weiss(2000), Kantznelson(2001), Bergelson+ Furstenberg+Weiss(2006), Host+Kra(2011), Huang+Ye(2002,2011)

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Furstenberg correspondence principle

Problem (Higher order problem) Let S ⊆ N be a syndetic set. How about the structure of the set {n ∈ N : S ∩ (S − n) ∩ (S − 2n) ∩ · · · ∩ (S − kn) = ∅} Similarly, we may consider the case that S has positive upper Banach density?

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Furstenberg correspondence principle

Using Furstenberg correspondence principle, the above problem is changed to the following dynamical problem: Problem (Higher order problem) Given a minimal system (X, T) and an non-empty open set U of X. How about the structure of the set {n ∈ N : U ∩ T−nU ∩ · · · ∩ T−knU = ∅}? RemarkµA dynamical system means a pair (X, T), where X is a compact metric space and T : X → X is a

homeomorphism. (X, T) is called minimal§if for each x ∈ X,
rb(x, T) = {Tnx : n ∈ Z} is dense in X.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Furstenberg correspondence principle–topological case

Proposition (Topological case) Let S ⊆ N be a syndetic set. Then there exist a minimal dynamical system(X, T) and a non-empty open subset U of X such that {n ∈ N : U ∩ T−nU ∩ · · · ∩ T−knU = ∅} ⊆ {n ∈ N : S ∩ (S − n) ∩ · · · ∩ (S − kn) = ∅}. For any minimal dynamical system(X, T) and non-empty open subset U of X, we can find a syndetic S ⊆ N such that {n ∈ N : S ∩ (S − n) ∩ · · · ∩ (S − kn) = ∅} ⊆ {n ∈ N : U ∩ T−nU ∩ · · · ∩ T−knU = ∅}.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Furstenberg correspondence principle–measurable case

Proposition (measurable case) Let S ⊆ N have positive upper Banach density. Then we can find a measure-preserving system (X, B, µ, T) and A ∈ B§µ(A) = BD∗(S) > 0 such that {n ∈ N : µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0} ⊆ {n ∈ N : S ∩ (S − n) ∩ · · · ∩ (S − kn) = ∅}. Let (X, B, µ, T) be a measure-preserving system and A ∈ B, µ(A) > 0. Then there exists S ⊆ N, BD∗(S) ≥ µ(A) such that {n ∈ N : S ∩ (S − n) ∩ · · · ∩ (S − kn) = ∅} ⊆ {n ∈ N : µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0}.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Measure-preserving system

RemarkµA measure-preserving system(X, B, µ, T) means (X, B, µ) is a probability space, T : X → X is a measure-preserving transformation, that is, T−1A ∈ B and µ(T−1A) = µ(A) for any A ∈ B.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Furstenberg correspondence principle

Higher order Problem: Let S ⊆ N have positive upper Banach density. How about the structure of the set Ck(S) := {n ∈ N : S ∩ (S − n) ∩ (S − 2n) ∩ · · · ∩ (S − kn) = ∅}? change to the following ergodic theory’s problem: Higher order Problem: Given a measure-preserving system (X, B, µ, T) and A ∈ B, µ(A) > 0. How about the structure of the Higher order return set {n ∈ N : µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0}? Furstenberg correspondence principle tell us: Szemeredi Theorem ⇐ ⇒The above set is non-empty.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple ergodic average

In 1977, Furstenberg proved thatµ Theorem (Furstenberg) Given a measure-preserving system (X, B, µ, T) and A ∈ B, µ(A) > 0. Then lim inf

N→+∞

1 N

N

n=1

µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0. Particularly, there exists n ∈ N such that µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0. ⇛Szemeredi Theorem.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple ergodic average

For learning the structure of the the Higher order return time set {n ∈ N : µ(A ∩ T−nA ∩ · · · ∩ T−knA) > 0}, we need to know more information about the average. For example, Is lim

N→+∞

1 N

N

n=1

µ(A ∩ T−nA ∩ · · · ∩ T−knA) existºect.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple ergodic average problem

The above question is generalized as the following:Multiple ergodic average problemµGiven a measure-preserving system (X, B, µ, T) and f1, f2, · · · , fk ∈ L∞(µ). Is lim

N→+∞

1 N

N

n=1

f1(Tnx)f2(T2nx) · · · fk(Tknx)

convergent in L2(µ)? (L2-ergodic average problem of order

k)

convergent for µ-a.e. x ∈ X? (Pointwise-ergodic average

problem of order k)

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple ergodic average problem

Furstenberg+Katznelson+Ornstein (1982) proved multiple L2-ergodic average Theorem for weakly mixing system. In later, we will see that in the study of Multiple ergodic average problem, there will naturally appear characteristic factor, nilsystem and nilsequence ect.. Using the higher order return time set and nilsequence, we will prove thata higher order return times set contains a alost Nil Bohr0 set, this is the reason of our conjecture.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple L2-ergodic average Theorem

L2-ergodic average Theorem of order 1⇐ = Von Neumann ergodic Theorem(1932). In 1984, Conze and Lesigne proved L2-ergodic average Theorem of order 2. In 2001, Host and Kra proved L2-ergodic average Theorem of order 3; Bergelson,Furstenberg+Weiss, Zhang ect. In 2005, Host and Kra proved Multiple L2-ergodic average Theorem(2007,Ziegler). Tao(2008),Austin(2010),Walsh(2011)....... Some survey on this topic: B.Kra (ICM2006), V.Bergelson (ICM2006), H.Furstenberg (ICM2010).

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Multiple pointwise-ergodic average Theorem

pointwise-ergodic average Theorem of order 1 ⇐ = Birkhoff ergodic Theorem(1931). In 1990, Bourgain proved pointwise-ergodic average Theorem of order 2. Huang, Shao and Ye (2014) proved pointwise-ergodic average Theorem of order k, k ≥ 3 for ergodic distal systems.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Characteristic factor and nilsystem

The Characteristic factor of a measure-preserving system (X, B, µ, T) was introduced by Furstenberg, Weiss in 1996, the key point lies inµ to show L2-ergodic average Theorem of order k for (X, B, µ, T) ⇐ ⇒ to show L2-ergodic average Theorem of

rder k for its k-Characteristic factor (Zk, Bk, µk, Tk).

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Characteristic factor and nilsystem

Using a kind of semi-norm (called Host-Kra- Gowers seminorm), Host and Kra(2005) provedµ Theorem (Host+Kra) k-Characteristic factor (Zk, Bk, µk, Tk) of a measure preserving (X, B, µ, T) is an inverse limit of k-nilsystem. For k-nilsystem, Multiple ergodic average Theorem had been investigated by Parry(1969), Lesigne(1989), Shah(1996), Leibman (2005), Ziegler (2005), See also Ratner equi-distribution Theorem(1991). This establisned the L2-ergodic average Theorem of order k.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Nilsequence

Definition (Nilsequence) Let (X, T) be a k-nilsystem, X = G/Γ, f : X → R is a continuous function. Then {f(Tnx)} is called a basic k-nilsequence"a k-nilsequence is uniformly limit of basic nilsequences. Theorem (Bergelson, Host and Kra, 2005) Let (X, B, µ, T) be an ergodic a measure preserving, k ∈ N and A ∈ B, µ(A) > 0. Then (µ(A ∩ T−nA ∩ · · · ∩ T−knA) = Fk(n) + a(n) where Fk is a k-nilsequence, a(n) convergences to zero in uniform density, i.e. lim sup

N→+∞

supM∈Z M+N−1

n=M

|a(n)| = 0.

SLIDE 30

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

A result

Using Bergelson-Host-Kra Theorem, Huang, Shao and Ye (Mem. Amer. Math. Soc.) prove the following resultµ Theorem Let (X, B, µ, T) be an ergodic measure-preserving system, k ∈ N and A ∈ B, µ(A) > 0. Then the set I = {n ∈ N : µ(A ∩ T−nA ∩ . . . ∩ T−knA) > 0} is an ”almost” Nilk Bohr0 set, that is, there exists M ⊂ N with BD∗(M) = 0 such that I ∪ M is a Nilk Bohr0 set. Remark: ”almost” Nilk Bohr0 set is syndetic.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

A result and related problem

Using Furstenberg correspondence principle (measurable case), we have Theorem (Huang,Shao+Ye, Mem. Amer. Math. Soc.) Let S ⊆ N have positive upper Banach density. Then {n ∈ N : S ∩ (S − n) ∩ (S − 2n) ∩ · · · ∩ (S − kn) = ∅} is an ”almost” Nilk Bohr0 set, particularly it is a syndetic set. RemarkµSince a syndetic set has positive positive upper Banach density. Hnece the above Theorem also holds for syndetic set.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Nilk Bohr0 set and generalized polynomials

Proposition (Huang,Shao+Ye, Mem. Amer. Math. Soc.) Let k ∈ N. Then a subset A of natural numbers is Nilk Bohr0 set if and only if there exist (special) generalized polynomials P1, P2, · · · , Pm of degree ≤ k and ǫ > 0 such that A ⊇

m

i=1

{n ∈ N : Pi(n)(mod Z) ∈ (−ǫ, ǫ)}.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Example of generalized polynomials

SGP1 = {an : a ∈ R}, SGP2 = {an2, bn⌈cn⌉, f⌈en2⌉ : a, b, c, e, f ∈ R}. a1⌈a2n2⌉ + b1⌈b2⌈b3n⌉⌉ + c1n2 + c2n ∈ GP2, where ai, bi, ci ∈ R. SGP3 = {an3, an2⌈bn⌉, an⌈bn2⌉, an⌈bn⌈cn⌉⌉ : a, b, c ∈ R} ∪ 2

i=1 SGPi.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

Remarks on generalized polynomials

Generalized polynomials have been studied extensively, For example: Bergelson and Leibman(2007, Acta Math) investigate the equi-distribution of Generalized polynomials, which is a generalization of A.Weil’s result on the equi-distribution mod1 of polynomials having at least a non-zero irration coefficient. Green and Tao(2012, Ann Math) proved that Generalized polynomials is asymptotic orthogonal to M¨

bius function.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

A related problem on prime numbers

Problem For any positive integer k ≥ 1, whether the set of the commom difference of arithmetic progressions of length k + 1 in primes is an ”almost” Nilk Bohr0 set? (or a syndetic?)

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
4. Maillet’s conjecture

Conjecture (Maillet,1905) Every even number is the difference of two primes. Actually, before Maillet’s conjecture, there were two stronger forms of the conjecture. Conjecture (Kronecker, 1901) Every even number can be expressed in infinitely many ways as the difference of two primes. Conjecture (Polignac, 1849) Every even number can be written in infinitely many ways as the difference of two consecutive primes.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
4. Zhang’s result

It is easy to see that The twin prime conjecture is a special case of Kronecker’s Conjecture. Recently, Zhang made a breakthrough and proved that Theorem There exists an even number not more than 7 × 107 which can be expressed in infinitely many ways as the difference

f two primes.

Soon after, Maynard and Tao reduced the limit of such even number to not more than 600. The best known result now is not more than 246.

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Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result
4. Zhang’s result

Let k be a positive integer, we say a given k-tuple of integers H = {h1, h2, · · · , hk} is admissible if for any prime p, hi’s never occupy all of the residue classes modulo p. Theorem (Zhang’s Theorem) Let H = {h1, h2, · · · , hk} be an k-tuple of integers. If H is admissible and k > C for some given constant C > 0, then there are infinitely many integers n such that at least two of the numbers n + h1, n + h2, · · · , n + hk will be prime. Zhang proved that C = 3.5 × 106, then obtained there are infinitely many couple of primes with difference not more than 7 × 107. To obtain Maynard and Tao’s result, they proved a much smaller value C = 105.

SLIDE 39

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

4.Pintz’s result

Based on the GPY sieve method’s result, Pintz proved Theorem Let ǫ > 0 be arbitrary. The interval [x, x + xǫ] contains even numbers which can be written as the difference of primes if x > x0(ǫ).

SLIDE 40

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result

4.Our main results

Theorem (Main result) Let D0={p − q: p > q and p, q are primes}, then D0 is syndetic. Actually, we prove a stronger form as Theorem (Main result) Let D={d: d can be expressed in infinitely many ways as the difference of two primes }, then D is syndetic. That is, There is L ∈ N such that D ∩ [m, m + L] = ∅ for any m ∈ N. The proof of Theorem is based on Zhang’s Theorem. However, our method doesn’t give the exact value of L.

SLIDE 41

Difference set

f primes and

related problems Wen Huang 1.Background 2.A conjecture

3. Higher
rder Bohr

problem

1).Furstenberg correspondence principle 2). Multiple ergodic average 3).A result

4. Main result