Green-Tao’s theorem on k prime tuples. Wang Yonghui Department of Mathematics Capital Normal University Beijing 10037 P.R. China [email protected] Abstract

1

What Green-Tao proved?

Furstenberg [1] proved the multiple ergodic theorem: Theorem 1 Let (X, X , µ, T ) be a measure preserving probability system, for any fixed postive integer k, and any set E ∈ X with µ (E) > 0, we have lim inf

N →∞

N  1 X  µ E ∩ T −n E ∩ · · · ∩ T −(k−1)n E > 0. N n=1

Therefore, there exists an integer n such that   µ E ∩ T −n E ∩ · · · ∩ T −(k−1)n E > 0. Green-Tao provided more explicit settings and self-contained proof, without assuming any assumption of ergodic theory. Denote by ZN = Z/N Z, and for a function f : ZN → C, the average of f is defined as: N 1 X E (f ) := f (n) . N n=1 They proved that [2, Theorem 3.5] Theorem 2 (Green-Tao’s Ergodic Theorem) Let ν : ZN → R+ be a kpseudorandom measure (15). Suppose f is a non-negative function satisfying that 0 ≤ f (x) ≤ ν (x) for all x ∈ ZN

1

and E (f ) ≥ δ for some δ > 0. Then we have for any fixed k ≥ 3, E (f (x) f (x + r) · · · f (x + (k − 1) r) | x, r ∈ ZN ) ≥ c (k, δ) , for sufficiently large N. Remark 1. Taking f = 1E the characteristic function of E, and ν = 1, we immediately get a quantitative version of Theorem 1. Remark 2. If one can prove that, for any fixed positive integers k, r, E (f (x) f (x + r) · · · f (x + (k − 1) r) | x ∈ ZN ) ≥ c (k, δ) . Then the k-tuple primes conjecture follows, which includes the twin prime conjecture. By choosing suitable ν and f related to the primes, they closed a famous arithmetic conjecture on k-tuple primes with the following theorem: Theorem 3 (Green-Tao’s arithmetic theorem) The set of prime numbers contain infinitely many arithmetic progressions of length k for all k. i.e. for any fixed k, there are infinitely many pairs (n, r) such that the series n, n + r, n + (k − 1) r are all primes. Indeed, their results holds for any subset of primes A, satifying that {A ∩ [1, N ]} lim > 0. N →∞ {primes ∩ [1, N ]} History of Green-Tao Theorem: 1. Hardy-Littlewood (1923), k-tuple conjecture; 2. van der Corput (1939), k = 3; 3. Szemer´edi (1975), A = N. Proof of Green-Tao’s arithmetic Theorem 3. It is sufficient to choose the appropriate function f, ν related to prime numbers, and prove that ν is a k-pseudorandom measure. Let  −1 −k−5 ˜ (n) if εk N ≤ n ≤ 2εk N , k 2 Λ f (n) = (1) 0 otherwise. ˜ (n) is the modified von Mangoldt function where Λ  Φ(W ) if W n + 1 is a prime, ˜ (n) = W log (W n + 1) Λ 0 otherwise. 2

Q where W = p≤log log N p. This “W -trick” removes the small primes from consideration, and makes the pseudorandom to be possible (more uniformly distributed among the arithmetic progressions). Hence, there exits a k-pseudorandom measure related to f : ( Φ(W ) Λ2R (W n+1) if εk N ≤ n ≤ 2εk N , W log R ν (n) := 1 otherwise. where R = R (N, k) and ΛR (n) =

X

µ (d) log

d|n d≤R

R . d

˜ (n) , ΛR (n) are modified from the usual von Mangoldt function Here, Λ  X n log p if n = pr , Λ (n) = µ (d) log = 0 otherwise. d d|n

which enable the tool of complex analysis by relating the generating function of P Λ (n) n−s to Riemann zeta-functions. Hence, by Green-Tao theorem 2, we have E (f (x) f (x + r) · · · f (x + (k − 1) r) | x, r ∈ ZN ) ≥ ck . The contribution from r = 0 is O( N1 logk N ) = o (1) . Therefore, there must exists integer r, n 6= 0 such that W n+1, W (n + r)+1, · · · , W (n + (k − 1) r)+1 are all primes. The theorem follows then. Further, we expect the multiple convergence: Conjecture 4 Let f be as in formula (1), then we have E (f (x) f (x + r) · · · f (x + (k − 1) r) | x, r ∈ ZN ) ∼ c (k) , as N → ∞. And then, we have # {(n, r) ∈ Pk |n ≤ N } ∼

c (k) N 2 , as N → ∞. logk N

where (n, r) ∈ Pk means that n + r, n + 2r, · · · , n + (k − 1) r are all primes. Understanding-1. This indicates that primes are “probablity” independent distribution (in arithmetic progression). As we know, a single prime in [1, N ] has the “probablity” log1k N , the k independent primes occurs as the probability 1 . The conjectured term then follows since the number of the choice of pair logk N (n, r) is of order N 2 .

3

Understanding-2. We can regard Green-Tao’s Theorem as the following prime Diophantine equations:  pk − pk−1 = r    ··· (A) p2 − p1 = r    p2 − p1 = r, and their result states that the number of solution (p1 , . . . , pk , r) for pi , r ≤ N 2 is greater than c (k) logNk N . As far as we know, before their ingenune method, one can only solve the type of prime Diophantine equations by circle method and sieve method.

2

Ergodic Theorems for proving Theorem

Szemer´edy has proved Theorem 2 when ν = νconst = 1 the constant function. Theorem 5 (Szemer´ edy) Let f : ZN → R be a nonnegative function, satisfying that 0 ≤ f (x) ≤ 1, for all x ∈ ZN , and E (f (x) |x ∈ ZN ) ≥ δ > 0. Then E (f (x + r) f (x + 2r) · · · f (x + (k − 1) r) |x, r ∈ ZN ) ≥ c (k, δ) . for sufficiently large N. Proof. Szemer´edy first proved it by Ergodic theory. Furstenberg, Gower, Tao also give different version of proof. see history in the explanation which follows proposition 2.3 of [2]. As a consequence, Szemer´edy immediately generate Corollary 6 For fixed 0 < δ ≤ 1, if A v N is any set such that A ∩ {1, . . . , N } is of cardinality of at least δN. Then A contains infinitely arithmetic progression of length k. For detecting prime numbers, the function f in formula 1 is of order log N which canot be bounded by constant measures. Insteadly, f is bounded by a k-pseudorandom measure ν, Green-Tao develop a strategy that: 1. Generate a σ-algebra B to mollify f by the conditional expectation function fU ⊥ (x) = E (f |B) (x) = E (f (y) |y ∈ B (x)) , where B (x) is the unique atom in B which contains x. It is obvious that fU ⊥ (x) = constant = E (f (x) |B0 ) = E (f (x) |x ∈ ZN ) ≤ 1 4

for B0 = {∅, ZN } . More subtle σ-algebra B0 , . . . , BK will be generated iteratively in the step 2, and every time, reserving the condition (i)

fU ⊥ (x) = fU ⊥ (x) = E (f |Bi ) (x) ≤ 1 + o (1) , i = 1, . . . , K. 2. After a finite iterative step, fU = f −fU ⊥ will be very small in the Gower’s uniform norms (see (2)), in another words kf − fU ⊥ kU k−1 = o (1) . Now Green-Tao’s Theorem 2 can be immediately obtained by Szemer´edy theorem and the following Generalized von-Neumann theorem. Theorem 7 (Generalized von Neumann Theorem) Let ν be a k-pseudorandom measure, and fi (x) ≤ ν (x) + 1 for all x ∈ ZN .Then we have   E (f0 (x + r) f1 (x + 2r) · · · fk−1 (x + (k − 1) r) |x, r ∈ ZN ) = O min kfj kU k−1 +o (1) . 0≤j≤k−1

Proof. This can be obtained by using Cauchy inequality k − 2 times and the bound fi (x) ≤ ν (x) + 1 for i = 1, . . . , k − 1. And hence, we obtain the result in combinational form of f0 (x) and ν (x) . The part of ν (x) will contribue 1 + o (1) by the linear form condition (the 1th/2 condion of k-pseudorandom). And then the final remaining part will take the form of f0 as 1/2k−1

  kf0 kU k−1 = E 

Y

 f0 (x + ωh) |x ∈ ZN , h ∈ Zk−1 N

.

(2)

ω∈{0,1}k−1

Using elementary method (although Fourier analyisis can also be interviewed), kf kU k−1 is proved to be a norm for k ≥ 3 by Gower, and hence named as Gowers uniformity norms. Proof of Green-Tao’s Ergodic Theorem. Let f = fU + fU ⊥ where the subscript U denotes the Gower’s uniformity. fU ⊥ = 1 + o (1), and kfU kU k−1 = o (1) . Then by Szemer´edy theorem E (fU ⊥ (x + r) fU ⊥ (x + 2r) · · · fU ⊥ (x + (k − 1) r) |x, r ∈ ZN ) ≥ c (k, δ) for sufficiently large N. Applying the generalized von Neumann theorem we thus see that E (f0 (x + r) f1 (x + 2r) · · · fk−1 (x + (k − 1) r) |x, r ∈ ZN ) = o (1) , whenever each fj (x) is equal to fU or fU ⊥ , with at least one fj equals to fU . Adding these two estimates together we obtain E (f (x + r) f (x + 2r) · · · f (x + (k − 1) r) |x, r ∈ ZN ) ≥ c (k, δ) − o (1) .

5

2.1

The strategy of getting fU ⊥ , fU

Let 0 < ε < 1 be a positive small number. We need to construct the σ-algebra B0 , . . . , BK iteratively, such that Target I. The k-pseudorandom measure ν reserves uniform-distribution with respect to BK as K goes large. Explicit speaking, there exist a BK -measurable set ΩK such that, (Prop 8.1 and in particular Prop 8.2), E ((ν + 1) 1ΩK ) = Oε (1) ;

k(1 − 1ΩK ) E ((ν − 1) |BK )kL∞ (ZN ) = Oε (1) .

Here, ΩK is added for technical reason (6). As a consequence, for 0 ≤ f ≤ ν, we immediately get kfU ⊥ kL∞ (ZN ) := k(1 − 1ΩK ) E (f |BK )kL∞ (ZN ) ≤ 1 + Oε (1) . for every iterative step. Target II. Energy increment happens as K goes large, that is, for FK+1 = (1 − 1ΩK ) (f − E (f |BK )) , k

if kFK+1 kU k−1 > ε1/2 , then we have

2


1 − 1Ω E (f |BK+1 ) L2 (Z K+1

N)

2

≥ k(1 − 1ΩK ) E (f |BK )kL2 (ZN ) + 2−2

2
Since 1 − 1ΩK+1 E (f |BK+1 ) L2 (Z

N)

k

+1

ε. (3) are always bounded by 1 + o (1) , after

2k

finite step K ≤ K0 = [2 /ε + 2], we must have k

kfU kU k−1 = kFK kU k−1 ≤ ε1/2 . Both Target I and Target II are attacked by introducing the Dual Function DF (9) into the iterative process. Let B0 , . . . , BK and F1 , F2 , . . . , FK+1 be defined as B0 = {∅, ZN } , Ω0 = ∅ F1 = (1 − 1Ω0 ) (f − E (f |B0 )) = f − E (f ) , ··· BK = B0 ∨ Bε,η (DF1 ) ∨ · · · ∨ Bε,η (DFK ) FK+1 = (1 − 1ΩK ) (f − E (f |BK )) ··· where ΩK+1 ⊇ ΩK is a small measurable set in BK+1 , which equals ΩK ∪ Ω. Indeed, the existence of Ω is promised by Proposition 7.3 that, for 0 < η < 21 .     E ((ν + 1) 1Ω ) = Oε η 1/2 ; k(1 − 1Ω ) E ((ν − 1) |BK )kL∞ (ZN ) = Oε η 1/2 . (4) 6

And so, uniform distribution of ν is consequently obtained for BK+1 , ΩK+1 . Proof of Target I (4), Prop 7.3. Each Bε,η (DFj ) can be generated by atoms of DFj−1 ([ε (n + α) , ε (n + α + 1)]) for some α in Prop7.2., with an immediate consequence that kDFj − E (DFj |BK )k ≤ ε. (5) Hence BK is generated by OK,ε (1) atoms by boundness of DFj (11). Let Ω be generated by the all small atoms A satisfied with E ((ν + 1) 1A ) ≤ η 1/2 .It
1/2 is apparent that E ((ν + 1) 1Ω ) = Oε η . For the second equation in (4), it suffices to prove that   E ((ν − 1) 1A ) = O η 1/2 (6) E (ν − 1|A) = E (1A ) for all atoms E ((ν + 1) 1A ) > η 1/2 . Since E ((ν − 1) 1A ) + 2E (1A ) = E ((ν + 1) 1A ) > η 1/2 , it will sufficient to prove that E ((ν − 1) 1A ) = O (η) . The triangle inequality is applied for this key formula. Explicit speaking, a continuous function ΨA is introduced so that E ((ν − 1) (1A − ΨA (DF1 , . . . , DFK ))) = O (η) .

(7)

This is obtained by an argument of pigeonhole principle and choosing 1A − Ψj (DFj ) to be a continous characteristic function of very small interval (Prop7.2). Now, we need the last involved formula that, for any continuous function Ψ, E ((ν − 1) Ψ (DF1 , . . . , DFK )) = O (η) , which is induced by (13) from correlation condition and by (16) from linear form condition of ν and the definition of Dual function. See (12) of Prop 6.2. k Proof of Target II (3) (Prop 8.2). If kFK+1 kU k−1 > ε1/2 , by definition of Dual function (10), hFK+1 , DFK+1 i = h(1 − 1ΩK ) (f − E (f |BK )) , DFK+1 i 2k−1

= kFK+1 kU k−1 > ε1/2 . From (5) and uniform distribution of ν w.r.t BK and BK+1 , we get by triangle inequality that  


 1 − 1ΩK+1 (f − E (f |BK )) , E (DFK+1 |BK+1 ) > ε1/2 − O η 1/2 − O (ε) .
Since 1 − 1ΩK+1 , E (f |BK ) and E (DFK+1 |BK+1 ) are all measurable in BK+1 , i.e. locally constant on the atoms of BK+1 . Hence by taking conditional expectation wrt BK+1 on the both side of the above inequality, we have  


 1 − 1ΩK+1 (E (f |BK+1 ) − E (f |BK )) , E (DFK+1 |BK+1 ) > ε1/2 −O η 1/2 −O (ε) . 7

By Cauchy inequality and the boundness of Dual function (11) of Lem 6.1, we have


k

1 − 1Ω (8) (E (f |BK+1 ) − E (f |BK )) L2 (Z ) > 2−2 +1 ε1/2 . K+1 N

This implies (3) thanks to Pythagoras’s theorem, by the approximate orthogonality



 1 − 1ΩK+1 E (f |BK ) , 1 − 1ΩK+1 (E (f |BK+1 ) − E (f |BK )) < η 1/2 . The left hand is majorized by

2E 1ΩK+1 − 1ΩK (f − E (f |BK )) ≤ 4E



1ΩK+1 − 1ΩK (ν + 1)


which is O η 1/2 by the uniform distribution property of ν.

3

Arithmetic Theorems for verifying k-pseudorandom measure ν.

This part due to the method of Goldston-Yildirim. To be continued.

4

4.1

Index

Dual Function

Proof. Define the dual function of F to be    DF (x) := E  

Y



  fω (x + ωh) |h ∈ Zk−1 N .  κ−1

(9)

ω∈{0,1} ω6=0k−1

Then we immediately get by the definition that (6.4) 2k−1

hF, DF i = kF kU k−1 .

(10)

Lemma 1 (Lemma 6.1 of [2], Boundness of Dual function) If |F | < ν+ 1, then dual function is bounded |DF | ≤ 22

k−1

by the linear form conditon of ν.

8

−1

+ o (1)

(11)

Proposition 8 (Propostion 6.2 in [2], Weakly mixing of Dual function) k−1 k−1 Set I = [−22 , 2
2 ], and let Φ : I k → R be a continous function in a compact subset E of C I k . For k-pseudorandom measure ν, if |Fi | < ν + 1 Then we have hν − 1, Φ (DF1 , . . . , DFK )i = oK,E (1) .

(12)

Proof. By Weierstrass approximation theorem, it is sufficient to prove for Φ = P the polynomial functions. Further, since the formula (16) kν − 1kU k−1 = o (1) holds by applying linear form conditions. Hence it is sufficient to prove that, D Y E f, DFj = OK (1) (13) for all f with kf kU k−1 ≤ 1.This formula follows by applying Gowers-CauchySchwarz inequality first, and the correlation condition (2th/2 condition of kpseudorandom, used only here in Tao’s paper) in the final step.

4.2

Argument of Pigeonhole Principle (Prop 7.2)

Proof of (7) Prop 7.2. For any bounded function G(= DF ), we have Z 1X
E 1G(x)∈[ε(n+α−η),ε(n+α+η)] (ν − 1) dα

(14)

0 n∈Z Z 1X

1 X 1G(x)∈[ε(n+α−η),ε(n+α+η)] (ν − 1) (m) dα 0 n∈Z N m∈Z N Z 1X 1 X = (ν − 1) (m) 1G(x)∈[ε(n+α−η),ε(n+α+η)] (m) dα N 0 =

m∈ZN

n∈Z

= 2ηE (ν − 1) , since X

 1G(x)∈[ε(n+α−η),ε(n+α+η)] (m) =

n∈Z

− η}, { G(m) + η}], 1, α + n0 ∈ [{ G(m) ε ε 0, otherwise.

By pigeonhole principle, there exists a α ∈ [0, 1] such that X
E 1G(x)∈[ε(n+α−η),ε(n+α+η)] (ν − 1) = O (η) . n∈Z

Therefore, we only need P to choose 1A − ΨA to be continous and has its support in the support of the n∈Z 1G(x)∈[ε(n+α−η),ε(n+α+η)] . 9

4.3

k-pseudorandom measure

The function ν : ZN → R+ is a measure if E (ν) = 1 + o (1) , as N → ∞.

(15)

A measure ν is k-pseudorandom if it satisfies • linear form condition • correllation conditon Linear form condition induce that — Generalized von-Neumann theorem; — The boundness of Dual function, (see application in (5) and (8)) — the smallness of Gower Norm of ν − 1,i.e. kν − 1kU d = o (1) ,

(16)

and then the weakly-mixing wrt Dual function Theorem 8. The correlation condition only has its application in the weakly-mixing Dual function Theorem 8.

References [1] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´edi on arithmetic progressions. J. d’Analyse Math., 31 (1977), 204– 256. MR0498471 (58:16583). 1 [2] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions. To appear, Ann. Math.arXiv:math.NT/0404188 v4 2 Aug 2005. 1, 4, 8, 9 [3] E. Szemer´edi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345.

10