From affine to two-source extractors via approximate duality Eli - - PowerPoint PPT Presentation

From affine to two-source extractors via approximate duality

Eli Ben-Sasson Noga Zewi

Computer Science Department Technion — Israel Institute of Technology

May 2011

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 1 / 17

Outline

1

Main points

2

Extractors, dispersers and bipartite Ramsey graphs

3

Description of results

4

Proof sketches

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 2 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi. Clearly D(A, B) = 1 iff A ⊆ b + B⊥ for some b ∈ Fn

2.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi. Clearly D(A, B) = 1 iff A ⊆ b + B⊥ for some b ∈ Fn

2.

Question: What if D(A, B) > 0.99 ?

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi. Clearly D(A, B) = 1 iff A ⊆ b + B⊥ for some b ∈ Fn

2.

Question: What if D(A, B) > 0.99 ? Answers (this talk):

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi. Clearly D(A, B) = 1 iff A ⊆ b + B⊥ for some b ∈ Fn

2.

Question: What if D(A, B) > 0.99 ? Answers (this talk):

1

There exist “large” subsets A′, B′ of A, B such that D(A′, B′) = 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

Approximate duality and discrepancy

Given A, B ⊂ Fn

2 let their duality constant (or discrepancy) be

D(A, B) =

• Ea∈A,b∈B
• (−1)a,b
• , where a, b =

n

• i=1

aibi. Clearly D(A, B) = 1 iff A ⊆ b + B⊥ for some b ∈ Fn

2.

Question: What if D(A, B) > 0.99 ? Answers (this talk):

1

There exist “large” subsets A′, B′ of A, B such that D(A′, B′) = 1.

2

Assuming the polynomial Freiman-Ruzsa conjecture (PFR), A′, B′ as above exist even when D(A, B) > 2−Ω(n).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 3 / 17

Main points

One main point

Theorem (Discrepancy in matrices of small rank)

Assuming PFR, for every α, δ > 0 exists γ > 0 such that: If M ∈ FN×N

2

has F2-rank at most log N

α

and discrepancy greater than 2−γn then M contains a large monochromatic rectangle M[S, T], |S|, |T| ≥ N1− δ

α . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 4 / 17

Main points

One main point

Theorem (Discrepancy in matrices of small rank)

Assuming PFR, for every α, δ > 0 exists γ > 0 such that: If M ∈ FN×N

2

has F2-rank at most log N

α

and discrepancy greater than 2−γn then M contains a large monochromatic rectangle M[S, T], |S|, |T| ≥ N1− δ

α .

Conjecture (Polynomial Freiman-Ruzsa (PFR))

Let σ2(A) = |A + A|/|A| where A + A = {a + a′ | a, a′ ∈ A}. There exists a constant c such that for every A ⊂ Fn

2 we have:

∃A′ ⊂ A, |A′| ≥ |A|/σ2(A)c such that |A′|/|span(A′)| ≥ σ2(A)−c.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 4 / 17

Extractors, dispersers and bipartite Ramsey graphs

Affine and two-source dispersers/extractors

Definition (Affine Extractor)

is a function f : Fn

2 → Fm 2 such that for all affine sources X ⊆ Fn 2 of

min-entropy rate dim(X)

n

at least ρ, f (X) − Um∞ ≤ ǫ.

Definition (Two-source extractor)

is a function f : Fn

2 × Fn 2 → Fm 2 such that for all pairs of independent

sources X, Y ⊆ Fn

2 of min-entropy rate at least ρ,

f (X, Y ) − Um∞ ≤ ǫ.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 5 / 17

Extractors, dispersers and bipartite Ramsey graphs

Affine and two-source dispersers/extractors

Definition (Affine Disperser)

is a function f : Fn

2 → F2 such that for all affine sources X ⊆ Fn 2 of

min-entropy rate dim(X)

n

at least ρ, |f (X)| > 1.

Definition (Two-source disperser)

is a function f : Fn

2 × Fn 2 → F2 such that for all pairs of independent

sources X, Y ⊆ Fn

2 of min-entropy rate at least ρ,

|f (X, Y )| > 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 5 / 17

Extractors, dispersers and bipartite Ramsey graphs

Three main points

“Good” affine extractors (small rate and error) can be converted in a black-box manner into “good” two-source dispersers (small rate). Two-source dispersers of low F2-rank are extractors with bounded error. Under the polynomial Freiman-Ruzsa conjecture (PFR), extractor error is exponentially small.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 6 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function BHRVW ‘01

1 2 + ǫ

exponentially small

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function BHRVW ‘01

1 2 + ǫ

exponentially small BKSSW ‘05 ǫ dispersers

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function BHRVW ‘01

1 2 + ǫ

exponentially small BKSSW ‘05 ǫ dispersers Bourgain ‘07 ǫ exponentially small

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function BHRVW ‘01

1 2 + ǫ

exponentially small BKSSW ‘05 ǫ dispersers Bourgain ‘07 ǫ exponentially small Yehudayoff ‘09, Li ‘10 ǫ/ log log n exponentially small

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

Previous constructions

Two-source Reference rate error Erd¨

• s ‘47 (prob. method)

O( log n

n )

disperser Chor, Goldreich ‘88

1 2 + ǫ

exponentially small Pudl´ ak, R¨

• dl ‘04

1 2 − O( 1 √n)

dispersers Bourgain ‘05

1 2 − ǫ0

exponentially small BKSSW ‘05 ǫ disperser BRSW ‘06 n1−ǫ disperser Affine Reference rate error Folklore (prob. method) O( log n

n )

random function BHRVW ‘01

1 2 + ǫ

exponentially small BKSSW ‘05 ǫ dispersers Bourgain ‘07 ǫ exponentially small Yehudayoff ‘09, Li ‘10 ǫ/ log log n exponentially small Ben-Sasson, Kopparty ‘09 n−3/5 disperser

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 7 / 17

Extractors, dispersers and bipartite Ramsey graphs

1

Main points

2

Extractors, dispersers and bipartite Ramsey graphs

3

Description of results

4

Proof sketches

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 8 / 17

Description of results

A pair of two-source extractors

Given f : Fn

2 → Fm 2 (presumably, an affine extractor)

The concatenated two-source extractor C : Fn

2 × Fn 2 → F2 is defined

by C(x, y) = x ◦ f (x), y ◦ f (y).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 9 / 17

Description of results

A pair of two-source extractors

Given f : Fn

2 → Fm 2 (presumably, an affine extractor)

The concatenated two-source extractor C : Fn

2 × Fn 2 → F2 is defined

by C(x, y) = x ◦ f (x), y ◦ f (y). Let F : Fn−m

2

→ Fn

2 be injective on f (−1)(z) for some z ∈ Fm 2 . The

preimage two-source extractor P : Fn−m

2

× Fn−m

2

→ F2 is defined by P(x, y) = F(x), F(y).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 9 / 17

Description of results

Main results — From affine to two-source

Assume f : Fn

2 → Fm 2 is an affine extractor for min entropy rate ρ with

ℓ∞-error at most 2−m.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 10 / 17

Description of results

Main results — From affine to two-source

Assume f : Fn

2 → Fm 2 is an affine extractor for min entropy rate ρ with

ℓ∞-error at most 2−m. The loss rate is defined as λ = 1 − m

ρn.

Smaller λ means more entropy is recovered by extractor.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 10 / 17

Description of results

Main results — From affine to two-source

Assume f : Fn

2 → Fm 2 is an affine extractor for min entropy rate ρ with

ℓ∞-error at most 2−m. The loss rate is defined as λ = 1 − m

ρn.

Smaller λ means more entropy is recovered by extractor. Bourgain ’07: Extractors for any ρ > 0, achieving λρ < 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 10 / 17

Description of results

Main results — From affine to two-source

Assume f : Fn

2 → Fm 2 is an affine extractor for min entropy rate ρ with

ℓ∞-error at most 2−m. The loss rate is defined as λ = 1 − m

ρn.

Smaller λ means more entropy is recovered by extractor. Bourgain ’07: Extractors for any ρ > 0, achieving λρ < 1.

Theorem (From affine extractors to two-source disperser)

The concatenated construction is a two-source disperser for min-entropy rate ρ′ satisfying (i) ρ′ < 1/2 when ρ < 1/2, and (ii) ρ′ λ→0 − → 2/5. The pre-image construction is a two-source disperser for rate ρ′ =

λ 1+λ λ→0

− → 0, as long as ρ ≤ 1/2.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 10 / 17

Description of results

Main results — rank, discrepancy, approximate duality

Given two-source function g : Fn

2 × Fn 2 → F2 let rank2(g) be the

F2-rank of Mg ∈ FFn

2×Fn 2

2

where Mg[x, y] = g(x, y). We have 0 ≤ rank2(g) ≤ 2n. Rank of previously shown constructions is linear in n.

Theorem (Low-rank dispersers are extractors)

Assume g is a two-source disperser of min-entropy rate ρ and rank O(n). g is a two-source extractor for min-entropy rate ρ + ǫ and error strictly less than 1

2.

Assuming PFR, g is a two-source extractor for min-entropy rate ρ + ǫ and error 2−Ω(ǫn).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 11 / 17

Description of results

Main results — rank, discrepancy, approximate duality

The doubling constant of A ⊂ Fn

2 is σ2(A) = |2A|/|A|, where

2A = {a + a′ | a, a′ ∈ A}. We have 1 ≤ σ2(A) ≤ |A| and σ2(A) = 1 iff A is a linear space.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 12 / 17

Description of results

Main results — rank, discrepancy, approximate duality

The doubling constant of A ⊂ Fn

2 is σ2(A) = |2A|/|A|, where

2A = {a + a′ | a, a′ ∈ A}. We have 1 ≤ σ2(A) ≤ |A| and σ2(A) = 1 iff A is a linear space.

Theorem (Freiman-Ruzsa)

If σ2(A) is small with respect to A then A contains a “large” subset that is a “large” fraction of a linear space. “Large” means exponential in σ2(A). PFR conjecture: “Large” means polynomial in σ2(A).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 12 / 17

Description of results

On PFR and approximate duality

Conjecture (PFR)

There exists a constant c such that every A ⊆ Fn

2 contains a subset A′ of

size |A|/(σ2(A))c such that |A′|/|span(A′)| ≥ σ2(A)c.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 13 / 17

Description of results

On PFR and approximate duality

Conjecture (PFR)

There exists a constant c such that every A ⊆ Fn

2 contains a subset A′ of

size |A|/(σ2(A))c such that |A′|/|span(A′)| ≥ σ2(A)c.

Conjecture (Approximate duality (ADC))

If A, B ⊂ Fn

2 are of size 2Ω(n) and have large discrepancy:

D(A, B) :=

• Ea∈A,b∈B
• (−1)a,b
• ≥ µ.

Then A, B contain large subsets that are contained in affine shifts of strictly dual sets.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 13 / 17

Description of results

On PFR and approximate duality

Conjecture (PFR)

There exists a constant c such that every A ⊆ Fn

2 contains a subset A′ of

size |A|/(σ2(A))c such that |A′|/|span(A′)| ≥ σ2(A)c.

Conjecture (Approximate duality (ADC))

If A, B ⊂ Fn

2 are of size 2Ω(n) and have large discrepancy:

D(A, B) :=

• Ea∈A,b∈B
• (−1)a,b
• ≥ µ.

Then A, B contain large subsets that are contained in affine shifts of strictly dual sets.

Theorem (PFR vs. ADC)

PFR implies ADC and ADC implies a weak (though unproven) form of PFR.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 13 / 17

Description of results

On PFR and ADC

Theorem (Approximate duality for nearly-dual sets)

For δ > 0 exists ǫ > 0 s.t. for A, B ⊂ Fn

2 with D(A, B) ≥ 1 − ǫ there exist

A′ ⊂ A, |A′| ≥ |A|/2 and B′ ⊂ B, |B′| ≥ 2−δn|B| with D(A′, B′) = 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 14 / 17

Description of results

On PFR and ADC

Theorem (Approximate duality for nearly-dual sets)

For δ > 0 exists ǫ > 0 s.t. for A, B ⊂ Fn

2 with D(A, B) ≥ 1 − ǫ there exist

A′ ⊂ A, |A′| ≥ |A|/2 and B′ ⊂ B, |B′| ≥ 2−δn|B| with D(A′, B′) = 1.

Theorem (Discrepancy in matrices of small rank)

Assuming PFR, for every α, δ > 0 exists γ > 0 such that the following

• holds. If M ∈ FN×N

2

has rank at most log N

α

and discrepancy greater than 2−γn then M contains a large monochromatic rectangle M[S, T], |S|, |T| ≥ N1− δ

α . Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 14 / 17

Description of results

1

Main points

2

Extractors, dispersers and bipartite Ramsey graphs

3

Description of results

4

Proof sketches

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 15 / 17

Proof sketches

From affine to two-source extractors

Theorem (From affine extractors to two-source disperser)

The concatenated construction is a two-source disperser for min-entropy rate ρ′ λ→0 − → 2/5. If ρ = 1/2 the pre-image construction is a two-source disperser for rate ρ′ =

λ 1+λ λ→0

− → 0.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 16 / 17

Proof sketches

From affine to two-source extractors

Theorem (From affine extractors to two-source disperser)

The concatenated construction is a two-source disperser for min-entropy rate ρ′ λ→0 − → 2/5. If ρ = 1/2 the pre-image construction is a two-source disperser for rate ρ′ =

λ 1+λ λ→0

− → 0.

Theorem (Low-rank dispersers are extractors)

Assume g is a two-source disperser of min-entropy rate ρ and rank O(n). g is a two-source disperser for min-entropy rate ρ + ǫ and error strictly less than 1

2.

Assuming PFR, g is a two-source extractor for min-entropy rate ρ + ǫ and error 2−Ω(ǫn).

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 16 / 17

Proof sketches

Approximate duality

Theorem (Approximate duality for nearly-dual sets)

For δ > 0 exists ǫ > 0 s.t. for A, B ⊂ Fn

2 with D(A, B) ≥ 1 − ǫ there exist

A′ ⊂ A, |A′| ≥ |A|/2 and B′ ⊂ B, |B′| ≥ 2−δn|B| with D(A′, B′) = 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 17 / 17

Proof sketches

Approximate duality

Theorem (Approximate duality for nearly-dual sets)

For δ > 0 exists ǫ > 0 s.t. for A, B ⊂ Fn

2 with D(A, B) ≥ 1 − ǫ there exist

A′ ⊂ A, |A′| ≥ |A|/2 and B′ ⊂ B, |B′| ≥ 2−δn|B| with D(A′, B′) = 1.

Theorem (PFR implies ADC)

Assuming PFR, for every α, δ > 0 exists γ > 0 s.t. the following holds. If A, B ⊂ Fn

2 are of size at least 2αn and have nontrivial discrepancy

D(A, B) ≥ 2−γn, then there exist A′ ⊂ A, B′ ⊂ B of size 2(α−δ)n such that D(A′, B′) = 1.

Eli Ben-Sasson, Noga Zewi (Technion) From affine to two-source extractors via approximate duality May 2011 17 / 17