The Green-Tao theorem and a relative Szemerédi theorem

Transcription

1 The Green-Tao theorem and a relative Szemerédi theorem Yufei Zhao Massachusetts Institute of Technology Based on joint work with David Conlon (Oxford) and Jacob Fox (MIT)

2 Green Tao Theorem (arxiv 2004; Annals of Math 2008) The primes contain arbitrarily long arithmetic progressions (AP).

3 Green Tao Theorem (arxiv 2004; Annals of Math 2008) The primes contain arbitrarily long arithmetic progressions (AP). Szemerédi s Theorem (1975) Every subset of N with positive density contains arbitrarily long APs. (upper) density of A N is lim sup [N] := {1, 2,..., N} N A [N] N

4 Green Tao Theorem (arxiv 2004; Annals of Math 2008) The primes contain arbitrarily long arithmetic progressions (AP). Szemerédi s Theorem (1975) Every subset of N with positive density contains arbitrarily long APs. (upper) density of A N is lim sup [N] := {1, 2,..., N} N A [N] N P = prime numbers Prime number theorem: P [N] N 1 log N

5 Proof strategy of Green Tao theorem N P = prime numbers P

6 Proof strategy of Green Tao theorem N P = prime numbers, S = almost primes P [N] P S with positive relative density, i.e., S [N] > δ S P

7 Proof strategy of Green Tao theorem N P = prime numbers, S = almost primes P [N] P S with positive relative density, i.e., S [N] > δ Step 1: S P Relative Szemerédi theorem (informally) If S N satisfies certain pseudorandomness conditions, then every subset of S with positive relative density contains long APs.

8 Proof strategy of Green Tao theorem N P = prime numbers, S = almost primes P [N] P S with positive relative density, i.e., S [N] > δ Step 1: S P Relative Szemerédi theorem (informally) If S N satisfies certain pseudorandomness conditions, then every subset of S with positive relative density contains long APs. Step 2: Construct a superset of primes that satisfies the pseudorandomness conditions. (Goldston Yıldırım sieve)

9 Relative Szemerédi theorem Relative Szemerédi theorem (informally) If S N satisfies certain pseudorandomness conditions, then every subset of S with positive relative density contains long APs. What pseudorandomness conditions? Green Tao: 1 Linear forms condition 2 Correlation condition

10 Relative Szemerédi theorem Relative Szemerédi theorem (informally) If S N satisfies certain pseudorandomness conditions, then every subset of S with positive relative density contains long APs. What pseudorandomness conditions? Green Tao: 1 Linear forms condition 2 Correlation condition A natural question (asked by Gowers & Green) Does relative Szemerédi theorem hold with weaker and more natural hypotheses?

11 Relative Szemerédi theorem Relative Szemerédi theorem (informally) If S N satisfies certain pseudorandomness conditions, then every subset of S with positive relative density contains long APs. What pseudorandomness conditions? Green Tao: 1 Linear forms condition 2 Correlation condition no longer needed A natural question (asked by Gowers & Green) Does relative Szemerédi theorem hold with weaker and more natural hypotheses? Theorem (Conlon, Fox, Z.) Yes! A weaker linear forms condition suffices.

12 Szemerédi s theorem Host set: N Relative Szemerédi theorem Host set: some sparse subset of integers Conclusion: relatively dense subsets contain long APs

13 Szemerédi s theorem Host set: N Relative Szemerédi theorem Host set: some sparse subset of integers Random host set Kohayakawa Luczak Rödl 96 3-AP, p N 1/2 Conlon Gowers 10+ Schacht 10+ k-ap, p N 1/(k 1) Pseudorandom host set Green Tao 08 Conlon Fox Z. 13+ linear forms + correlation linear forms Conclusion: relatively dense subsets contain long APs

14 Roth s theorem Roth s theorem (1952) If A [N] is 3-AP-free, then A = o(n). [N] := {1, 2,..., N} 3-AP = 3-term arithmetic progression

15 Roth s theorem Roth s theorem (1952) If A [N] is 3-AP-free, then A = o(n). [N] := {1, 2,..., N} 3-AP = 3-term arithmetic progression It ll be easier (and equivalent) to work in Z N := Z/NZ.

16 Roth s theorem Roth s theorem (1952) If A [N] is 3-AP-free, then A = o(n). [N] := {1, 2,..., N} 3-AP = 3-term arithmetic progression It ll be easier (and equivalent) to work in Z N := Z/NZ. Roth s original proof uses Fourier analysis. Let us recall a graph theoretic proof.

17 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. G A X Y Z

18 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. G A x y iff 2x + y A X x Y y Z

19 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). X Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. G A x x z iff x z A Y z Z

20 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. G A X Y y y z iff y 2z A z Z

21 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). X Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. G A x y iff 2x + y A x x z iff x z A Y y y z iff y 2z A z Z

22 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. Triangle xyz in G A 2x + y, x z, y 2z A G A x y iff 2x + y A Y y X x y z iff y 2z A x z iff x z A z Z

23 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. Triangle xyz in G A 2x + y, x z, y 2z A It s a 3-AP with diff x y z G A x y iff 2x + y A Y y X x y z iff y 2z A x z iff x z A z Z

24 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. Triangle xyz in G A 2x + y, x z, y 2z A It s a 3-AP with diff x y z No triangles? Y G A x y iff 2x + y A y X x y z iff y 2z A x z iff x z A z Z

25 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. Triangle xyz in G A 2x + y, x z, y 2z A y It s a 3-AP with diff x y z Y y z iff y 2z A No triangles? Only triangles trivial 3-APs with diff 0. G A x y iff 2x + y A X x x z iff x z A z Z

26 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Given A, construct tripartite graph G A with vertex sets X = Y = Z = Z N. Triangle xyz in G A 2x + y, x z, y 2z A y It s a 3-AP with diff x y z Y y z iff y 2z A No triangles? Only triangles trivial 3-APs with diff 0. Every edge of the graph is contained in exactly one triangle (the one with x + y + z = 0). G A x y iff 2x + y A X x x z iff x z A z Z

27 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Constructed a graph with 3N vertices 3N A edges every edge lies in exactly one triangle Y G A x y iff 2x + y A y X x y z iff y 2z A x z iff x z A z Z

28 Proof of Roth s theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Constructed a graph with 3N vertices 3N A edges every edge lies in exactly one triangle Theorem (Ruzsa & Szemerédi 76) Y G A x y iff 2x + y A y X x y z iff y 2z A If every edge in a graph G = (V, E) is contained in exactly one triangle, then E = o( V 2 ). (a consequence of the triangle removal lemma) So 3N A = o(n 2 ). Thus A = o(n). x z iff x z A z Z

29 Relative Roth theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Relative Roth theorem (Conlon, Fox, Z.) If S Z N satisfies the 3-linear forms condition, and A S is 3-AP-free, then A = o( S ).

30 Relative Roth theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Relative Roth theorem (Conlon, Fox, Z.) If S Z N satisfies the 3-linear forms condition, and A S is 3-AP-free, then A = o( S ). G S x y iff 2x + y S Z N x x z iff x z S Z N y y z iff y 2z S z Z N

31 Relative Roth theorem Roth s theorem (1952) If A Z N is 3-AP-free, then A = o(n). Relative Roth theorem (Conlon, Fox, Z.) If S Z N satisfies the 3-linear forms condition, and A S is 3-AP-free, then A = o( S ). G S x y iff 2x + y S Z N x x z iff x z S 3-linear forms condition: G S has asymp. the same H-density as a random graph for every H K 2,2,2 Z N y y z iff y 2z S z Z N

32 Analogy with quasirandom graphs Chung-Graham-Wilson 89 showed that in constant edge-density graphs, many quasirandomness conditions are equivalent, one of which is having the correct C 4 count

33 Analogy with quasirandom graphs Chung-Graham-Wilson 89 showed that in constant edge-density graphs, many quasirandomness conditions are equivalent, one of which is having the correct C 4 count In sparse graphs, the Chung Graham Wilson equivalences do not hold.

34 Analogy with quasirandom graphs Chung-Graham-Wilson 89 showed that in constant edge-density graphs, many quasirandomness conditions are equivalent, one of which is having the correct C 4 count 2-blow-up In sparse graphs, the Chung Graham Wilson equivalences do not hold. Our results can be viewed as saying that: Many extremal and Ramsey results about H (e.g., H = K 3 ) in sparse graphs hold if there is a host graph that behaves pseudorandomly with respect to counts of the 2-blow-up of H. 2-blow-up

35 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ).

36 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y

37 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y 4-linear forms condition: H-density in this hypergraph is asymp. same as random for any H K (3) (3) 4 [2] := 2-blow-up of K 4

38 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y 4-linear forms condition: H-density in this hypergraph is asymp. same as random for any H K (3) (3) 4 [2] := 2-blow-up of K 4

39 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y 4-linear forms condition: H-density in this hypergraph is asymp. same as random for any H K (3) (3) 4 [2] := 2-blow-up of K 4

40 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y 4-linear forms condition: H-density in this hypergraph is asymp. same as random for any H K (3) (3) 4 [2] := 2-blow-up of K 4

41 Relative Szemerédi theorem (Conlon, Fox, Z.) Fix k 3. If S Z N satisfies the k-linear forms condition, and A S is k-ap-free, then A = o( S ). k = 4: build a 4-partite 3-uniform hypergraph Vertex sets W = X = Y = Z = Z N wxy E 3w + 2x + y S wxz E 2w + x z S wyz E w y 2z S xyz E x 2y 3z S 4-AP with common diff: w x y z W w x X Z z y Y 4-linear forms condition: H-density in this hypergraph is asymp. same as random for any H K (3) (3) 4 [2] := 2-blow-up of K 4

42 Roth s theorem: from one 3-AP to many 3-APs Roth s theorem δ > 0, for sufficiently large N, every A Z N with A δn contains a 3-AP.

43 Roth s theorem: from one 3-AP to many 3-APs Roth s theorem δ > 0, for sufficiently large N, every A Z N with A δn contains a 3-AP. By an averaging argument (Varnavides), we get many 3-APs: Roth s theorem (counting version) δ > 0 c > 0 so that for sufficiently large N, every A Z N with A δn contains at least cn 2 many 3-APs.

44 Transference Start with (sparse) A S Z N, A δ S

45 Transference Start with (sparse) A S Z N, A δ S One can find a dense model à for A: (dense) à Z N, à N A S δ

46 Transference Start with (sparse) A S Z N, A δ S One can find a dense model à for A: (dense) à Z N, à N A S δ Counting lemma will tell us that ( ) N 3 {3-APs in A} {3-APs in S Ã}

47 Transference Start with (sparse) A S Z N, A δ S One can find a dense model à for A: (dense) à Z N, à N A S δ Counting lemma will tell us that ( ) N 3 {3-APs in A} {3-APs in S Ã} cn 2 [By Roth s Theorem] (blackbox application) = relative Roth theorem (also works for k-term AP)

48 Converting to functional language Roth s theorem (counting version) δ > 0 c > 0 so that for sufficiently large N, every A Z N with A δn contains at least cn 2 many 3-APs.

49 Converting to functional language Roth s theorem (counting version) δ > 0 c > 0 so that for sufficiently large N, every A Z N with A δn contains at least cn 2 many 3-APs. Roth s theorem (weighted version) δ > 0 c > 0 so that for sufficiently large N, every f : Z N [0, 1] with Ef δ satisfies AP 3 (f ) := E x,d ZN [f (x)f (x + d)f (x + 2d)] c.

50 Sparse setting Sparse set A S Z N correspond to (normalized) indicator functions A δ S becomes Ef δ. ν = N S 1 S and f = N S 1 A.

51 Sparse setting Sparse set A S Z N correspond to (normalized) indicator functions A δ S becomes Ef δ. ν = N S 1 S and f = N S 1 A. More generally, we consider any (say that f is majorized by ν) f ν : Z N [0, ) (pointwise inequality) with Eν = 1 and Ef δ.

52 Roth s theorem (weighted version) δ > 0 c > 0 so that for sufficiently large N, every f : Z N [0, 1] with Ef δ satisfies AP 3 (f ) c.

53 Roth s theorem (weighted version) δ > 0 c > 0 so that for sufficiently large N, every f : Z N [0, 1] with Ef δ satisfies AP 3 (f ) c. Relative Roth theorem (Conlon, Fox, Z.) δ > 0 c > 0 so that for sufficiently large N, if ν : Z N [0, ) satisfies the 3-linear forms condition, and f : Z N [0, ) majorized by ν and Ef δ, then AP 3 (f ) c. Recall AP 3 (f ) = E x,d ZN [f (x)f (x + d)f (x + 2d)] Remark. The dependence of c on δ is the same.

54 3-linear forms condition The density of K 2,2,2 in Z N x ν(2x + y) ν(x z) y z Z N ν( y 2z) Z N

55 Relative Roth theorem (Conlon, Fox, Z.) δ > 0 c > 0 so that for sufficiently large N, if ν : Z N [0, ) satisfies the 3-linear forms condition, and f : Z N [0, ) majorized by ν and Ef δ, then AP 3 (f ) c. ν : Z N [0, ) satisfies the 3-linear forms condition if E[ν(2x + y)ν(2x + y)ν(2x + y )ν(2x + y ) ν(x z)ν(x z)ν(x z )ν(x z ) ν( y 2z)ν( y 2z)ν( y 2z )ν( y 2z )] = 1 + o(1) as well as if any subset of the 12 factors were deleted.

56 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ

57 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef

58 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f )

59 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f ) c [By Roth s Thm (weighted version)] = relative Roth theorem

60 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f ) c [By Roth s Thm (weighted version)] = relative Roth theorem

61 In what sense does 0 f 1 approximate 0 f ν?

62 In what sense does 0 f 1 approximate 0 f ν? Green Tao: based on Gowers uniformity norm Our approach: cut norm (aka discrepancy)

63 In what sense does 0 f 1 approximate 0 f ν? Green Tao: based on Gowers uniformity norm Our approach: cut norm (aka discrepancy) Using cut norm: Cheaper dense model theorem More difficult counting lemma

64 Cut norm for weighted bipartite graph (Frieze Kannan): g : X Y R 1 g := X Y sup X g(x, y) A X B Y A x A y B Y B

65 Cut norm for weighted bipartite graph (Frieze Kannan): g : X Y R 1 g := X Y sup X g(x, y) A X B Y A Cut norm for Z N : f : Z N R f := 1 N 2 x A y B sup A,B Z N f (x + y) x A y B Y B

66 Cut norm for weighted bipartite graph (Frieze Kannan): g : X Y R 1 g := X Y sup X g(x, y) A X B Y A Cut norm for Z N : f : Z N R f := 1 N 2 x A y B sup A,B Z N f (x + y) x A y B Y B Dense model theorem Assume ν : Z N [0, ) satisfies ν 1 = o(1). Then 0 f ν, f : Z N [0, 1] s.t. f f = o(1).

67 Dense model theorem Dense model theorem Assume ν : Z N [0, ) satisfies ν 1 = o(1). Then 0 f ν, f : Z N [0, 1] s.t. f f = o(1).

68 Dense model theorem Dense model theorem Assume ν : Z N [0, ) satisfies ν 1 = o(1). Then 0 f ν, f : Z N [0, 1] s.t. f f = o(1). Proof of the general dense model theorem 1. Regularity-type energy-increment argument (Green Tao, Tao Ziegler) 2. Separating hyperplane theorem (Hahn-Banach) + Weierstrass polynomial approximation theorem (Gowers & Reingold Trevisan Tulsiani Vadhan)

69 Dense model theorem Dense model theorem Assume ν : Z N [0, ) satisfies ν 1 = o(1). Then 0 f ν, f : Z N [0, 1] s.t. f f = o(1). Proof of the general dense model theorem 1. Regularity-type energy-increment argument (Green Tao, Tao Ziegler) 2. Separating hyperplane theorem (Hahn-Banach) + Weierstrass polynomial approximation theorem (Gowers & Reingold Trevisan Tulsiani Vadhan) Specialized/simplified for the cut norm on Z N (Z.)

70 Higher cut norms (for 4-term AP) 3-uniform weighted hypergraph g : X Y Z R, define 1 g := g(x, y, z) X Y Z. sup A Y Z B X Z C X Y x X,y Y,z Z (y,z) A (x,z) B (x,y) C i.e., supremum taken over all 2-graphs between X, Y, Z For f : Z N R, f,3 := sup E x,y,z Z N f (x + y + z)a(y, z)b(x, z)c(x, y) a,b,c : Z N [0,1]

71 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f ) c [By Roth s Thm (weighted version)] = relative Roth theorem

72 Transference Start with f ν : Z N [0, ) (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f ) c [By Roth s Thm (weighted version)] = relative Roth theorem

73 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z

74 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))1 A (x)1 B (y)] ɛ A X, B Y

75 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1]

76 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1] t(g) = E[g(x, y)g(x, z)g(y, z)]

77 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1] t(g) = E[g(x, y)g(x, z)g(y, z)] = E[ g(x, y)g(x, z)g(y, z)] + O(ɛ)

78 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1] t(g) = E[g(x, y)g(x, z)g(y, z)] = E[ g(x, y)g(x, z)g(y, z)] + O(ɛ) = E[ g(x, y) g(x, z)g(y, z)] + O(ɛ)

79 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1] t(g) = E[g(x, y)g(x, z)g(y, z)] = E[ g(x, y)g(x, z)g(y, z)] + O(ɛ) = E[ g(x, y) g(x, z)g(y, z)] + O(ɛ) = E[ g(x, y) g(x, z) g(y, z)] + O(ɛ) = t( g) + O(ɛ)

80 Counting lemma Weighted graphs g, g : (X Y ) (X Z) (Y Z) R Triangle density t(g) := E x,y,z [g(x, y)g(x, z)g(y, z)] Triangle counting lemma (dense setting) Assume 0 g, g 1. If g g ɛ, then t(g) = t( g) + O(ɛ). y x z E[(g(x, y) g(x, y))a(x)b(y)] ɛ a: X [0, 1], b : Y [0, 1] t(g) = E[g(x, y)g(x, z)g(y, z)] = E[ g(x, y)g(x, z)g(y, z)] + O(ɛ) = E[ g(x, y) g(x, z)g(y, z)] + O(ɛ) = E[ g(x, y) g(x, z) g(y, z)] + O(ɛ) = t( g) + O(ɛ) This argument doesn t work in the sparse setting (g unbounded)

81 Sparse counting lemma Sparse triangle counting lemma (Conlon, Fox, Z.) Assume that ν satisfies the 3-linear forms condition. If 0 g ν, 0 g 1 and g g = o(1), then Recall t(g) = E[g(x, y)g(x, z)g(y, z)] t(g) = t( g) + o(1)

82 Sparse counting lemma Sparse triangle counting lemma (Conlon, Fox, Z.) Assume that ν satisfies the 3-linear forms condition. If 0 g ν, 0 g 1 and g g = o(1), then Recall t(g) = E[g(x, y)g(x, z)g(y, z)] Proof ingredients 1 Cauchy-Schwarz 2 Densification t(g) = t( g) + o(1) 3 Apply cut norm/discrepancy (as in dense case)

83 Densification x E[g(x, z)g(y, z)g(x, z )g(y, z )] y z z

84 Densification x E[g(x, z)g(y, z)g(x, z )g(y, z )] y z z Set g (x, y) := E z [g(x, z )g(y, z )], i.e., normalized codegrees g (x, y) 1 for almost all (x, y) (since g ν and ν is pseudorandom) g behaves like a dense weighted graph

85 Densification x E[g(x, z)g(y, z)g(x, z )g(y, z )] = E[g (x, y)g(x, z)g(y, z)] y z Set g (x, y) := E z [g(x, z )g(y, z )], i.e., normalized codegrees g (x, y) 1 for almost all (x, y) (since g ν and ν is pseudorandom) g behaves like a dense weighted graph

86 Densification x E[g(x, z)g(y, z)g(x, z )g(y, z )] = E[g (x, y)g(x, z)g(y, z)] y z Set g (x, y) := E z [g(x, z )g(y, z )], i.e., normalized codegrees g (x, y) 1 for almost all (x, y) (since g ν and ν is pseudorandom) g behaves like a dense weighted graph Made X Y dense. Now repeat for X Z and Y Z. Reduce to dense setting.

87 Transference Start with f ν (sparse) f : Z N [0, ) Ef δ Dense model theorem: one can approximate f (in cut norm) by (dense) f : ZN [0, 1] E f = Ef Counting lemma implies AP 3 (f ) AP 3 ( f ) c [By Roth s Thm (weighted version)] = relative Roth theorem

88 Further application of densification Tao & Ziegler (arxiv Sept 2014): Narrow progressions in the primes

89 Further application of densification Tao & Ziegler (arxiv Sept 2014): Narrow progressions in the primes Any δ-proportion of primes N contains a k-ap with common difference O(log L k N)

90 Further application of densification Tao & Ziegler (arxiv Sept 2014): Narrow progressions in the primes Any δ-proportion of primes N contains a k-ap with common difference O(log L k N) Also for polynomial progression in the primes (originally Tao Ziegler 08)

91 Further application of densification Tao & Ziegler (arxiv Sept 2014): Narrow progressions in the primes Any δ-proportion of primes N contains a k-ap with common difference O(log L k N) Also for polynomial progression in the primes (originally Tao Ziegler 08) Proof uses densification.

92 Further application of densification Tao & Ziegler (arxiv Sept 2014): Narrow progressions in the primes Any δ-proportion of primes N contains a k-ap with common difference O(log L k N) Also for polynomial progression in the primes (originally Tao Ziegler 08) Proof uses densification. Open Problem (bounded gaps) Prove there exist infinitely many 3-APs of primes with bounded common difference. Maynard/Tao: infinitely many intervals of length k with log k primes.

93 Further remarks The transference proof applies Szemerédi s theorem as a black box to the sparse relative setting (preserving quantitative bounds).

94 Further remarks The transference proof applies Szemerédi s theorem as a black box to the sparse relative setting (preserving quantitative bounds). Same applies to multidimensional Szemerédi theorem: Theorem (Tao 06) The Gaussian primes contain arbitrary constellations.

95 Further remarks The transference proof applies Szemerédi s theorem as a black box to the sparse relative setting (preserving quantitative bounds). Same applies to multidimensional Szemerédi theorem: Theorem (Tao 06) The Gaussian primes contain arbitrary constellations. The situation for dense subsets of P P is quite different. See Tao Ziegler & Cook Magyar Titichetrakun (also Fox Z.)

96 Relative hypergraph removal lemma More generally, we transfer the hypergraph removal lemma to the sparse relative setting.

97 Relative hypergraph removal lemma More generally, we transfer the hypergraph removal lemma to the sparse relative setting. E.g., (everything generalizes to hypergraphs) Triangle removal lemma If G is a graph on N vertices with o(n 3 ) triangles, then all triangles can be removed by deleting o(n 2 ) edges.

98 Relative hypergraph removal lemma More generally, we transfer the hypergraph removal lemma to the sparse relative setting. E.g., (everything generalizes to hypergraphs) Triangle removal lemma If G is a graph on N vertices with o(n 3 ) triangles, then all triangles can be removed by deleting o(n 2 ) edges. Relative triangle removal lemma (Conlon, Fox, Z.) Let Γ be a graph on N vertices and edge-density p satisfying the triangle-linear forms condition, and G a subgraph of Γ. If G has o(p 3 N 3 ) triangles, then all triangles of G can be removed by deleting o(pn 2 ) edges. The triangle-linear forms condition is the pseudorandomness w.r.t. H-density, whenever H K 2,2,2 (as we saw earlier).

99 Relative hypergraph removal lemma More generally, we transfer the hypergraph removal lemma to the sparse relative setting. E.g., (everything generalizes to hypergraphs) Triangle removal lemma If G is a graph on N vertices with o(n 3 ) triangles, then all triangles can be removed by deleting o(n 2 ) edges. Relative triangle removal lemma (Conlon, Fox, Z.) Let Γ be a graph on N vertices and edge-density p satisfying the triangle-linear forms condition, and G a subgraph of Γ. If G has o(p 3 N 3 ) triangles, then all triangles of G can be removed by deleting o(pn 2 ) edges. The triangle-linear forms condition is the pseudorandomness w.r.t. H-density, whenever H K 2,2,2 (as we saw earlier). This gives another route for proving the relative Szemerédi theorem.

100 References Conlon, Fox, Zhao A relative Szemerédi theorem, 20pp Zhao An arithmetic transference proof of a relative Szemerédi thm, 6pp Conlon, Fox, Zhao The Green-Tao theorem: an exposition, 26pp