Jacob Fox Stanford University Matematick e Kolokvium 99...
Transcript of Jacob Fox Stanford University Matematick e Kolokvium 99...
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Arithmetic regularity, removal, and progressions
Jacob Fox
Stanford University
Matematicke Kolokvium 99
Prague
November 22, 2016
![Page 2: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/2.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 3: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/3.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 4: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/4.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 5: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/5.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 6: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/6.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 7: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/7.jpg)
Roth’s theorem
Theorem (Roth)
Every subset A ⊂ [N] with no three-term arithmetic progression
has |A| = o(N).
Roth: |A| = O(N/ log logN).
Improvements by Heath-Brown, Szemeredi, Bourgain.
Best known: |A| ≤ N/(logN)1−o(1) by Sanders, Bloom.
Behrend construction gives a lower bound of Nec√
log N.
![Page 8: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/8.jpg)
Game of Set
81 cards corresponding to points in F43.
Question
How many cards can we have without a “set”?
Answer: 20
![Page 9: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/9.jpg)
Game of Set
81 cards corresponding to points in F43.
Question
How many cards can we have without a “set”?
Answer: 20
![Page 10: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/10.jpg)
Game of Set
81 cards corresponding to points in F43.
Question
How many cards can we have without a “set”?
Answer: 20
![Page 11: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/11.jpg)
Game of Set
81 cards corresponding to points in F43.
Question
How many cards can we have without a “set”?
Answer: 20
![Page 12: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/12.jpg)
Cap Set Problem
Question
How large can A ⊂ Fn3 be without a 3-term arithmetic progression?
This variant of Roth’s theorem is related to several famousproblems in combinatorics and computer science, including thematrix multiplication problem, the sunflower conjecture.
Brown-Buhler: |A| = o(N).
Meshulam: |A| = O(3n/n).
Bateman-Katz: |A| = O(3n/n1+c).
![Page 13: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/13.jpg)
Cap Set Problem
Question
How large can A ⊂ Fn3 be without a 3-term arithmetic progression?
This variant of Roth’s theorem is related to several famousproblems in combinatorics and computer science, including thematrix multiplication problem, the sunflower conjecture.
Brown-Buhler: |A| = o(N).
Meshulam: |A| = O(3n/n).
Bateman-Katz: |A| = O(3n/n1+c).
![Page 14: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/14.jpg)
Cap Set Problem
Question
How large can A ⊂ Fn3 be without a 3-term arithmetic progression?
This variant of Roth’s theorem is related to several famousproblems in combinatorics and computer science, including thematrix multiplication problem, the sunflower conjecture.
Brown-Buhler: |A| = o(N).
Meshulam: |A| = O(3n/n).
Bateman-Katz: |A| = O(3n/n1+c).
![Page 15: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/15.jpg)
Cap Set Problem
Question
How large can A ⊂ Fn3 be without a 3-term arithmetic progression?
This variant of Roth’s theorem is related to several famousproblems in combinatorics and computer science, including thematrix multiplication problem, the sunflower conjecture.
Brown-Buhler: |A| = o(N).
Meshulam: |A| = O(3n/n).
Bateman-Katz: |A| = O(3n/n1+c).
![Page 16: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/16.jpg)
Cap Set Problem
Question
How large can A ⊂ Fn3 be without a 3-term arithmetic progression?
This variant of Roth’s theorem is related to several famousproblems in combinatorics and computer science, including thematrix multiplication problem, the sunflower conjecture.
Brown-Buhler: |A| = o(N).
Meshulam: |A| = O(3n/n).
Bateman-Katz: |A| = O(3n/n1+c).
![Page 17: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/17.jpg)
Breakthrough
Theorem (Croot, Lev, Pach)
If A ⊂ Zn4 has no 3-AP, then |A| ≤ 4cn with c ≈ .926.
Theorem (Ellenberg, Gijswijt)
If A ⊂ Fnp has no 3-AP, then |A| ≤ p(1−cp)n for an explicit cp > 0.
Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, Alon
Same conclusion for the multicolored sum-free problem: If
{xi}mi=1,{yi}mi=1,{zi}mi=1 ⊂ Fnp with xi + yj + zk = 0⇔ i = j = k ,
then m ≤ p(1−cp)n.
Theorem
Exponent is sharp for the multicolored sum-free problem: for
F2 by construction of Fu-Kleinberg, Fp by Kleinberg-Sawin-Speyer.
![Page 18: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/18.jpg)
Breakthrough
Theorem (Croot, Lev, Pach)
If A ⊂ Zn4 has no 3-AP, then |A| ≤ 4cn with c ≈ .926.
Theorem (Ellenberg, Gijswijt)
If A ⊂ Fnp has no 3-AP, then |A| ≤ p(1−cp)n for an explicit cp > 0.
Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, Alon
Same conclusion for the multicolored sum-free problem: If
{xi}mi=1,{yi}mi=1,{zi}mi=1 ⊂ Fnp with xi + yj + zk = 0⇔ i = j = k ,
then m ≤ p(1−cp)n.
Theorem
Exponent is sharp for the multicolored sum-free problem: for
F2 by construction of Fu-Kleinberg, Fp by Kleinberg-Sawin-Speyer.
![Page 19: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/19.jpg)
Breakthrough
Theorem (Croot, Lev, Pach)
If A ⊂ Zn4 has no 3-AP, then |A| ≤ 4cn with c ≈ .926.
Theorem (Ellenberg, Gijswijt)
If A ⊂ Fnp has no 3-AP, then |A| ≤ p(1−cp)n for an explicit cp > 0.
Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, Alon
Same conclusion for the multicolored sum-free problem:
If
{xi}mi=1,{yi}mi=1,{zi}mi=1 ⊂ Fnp with xi + yj + zk = 0⇔ i = j = k ,
then m ≤ p(1−cp)n.
Theorem
Exponent is sharp for the multicolored sum-free problem: for
F2 by construction of Fu-Kleinberg, Fp by Kleinberg-Sawin-Speyer.
![Page 20: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/20.jpg)
Breakthrough
Theorem (Croot, Lev, Pach)
If A ⊂ Zn4 has no 3-AP, then |A| ≤ 4cn with c ≈ .926.
Theorem (Ellenberg, Gijswijt)
If A ⊂ Fnp has no 3-AP, then |A| ≤ p(1−cp)n for an explicit cp > 0.
Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, Alon
Same conclusion for the multicolored sum-free problem: If
{xi}mi=1,{yi}mi=1,{zi}mi=1 ⊂ Fnp with xi + yj + zk = 0⇔ i = j = k ,
then m ≤ p(1−cp)n.
Theorem
Exponent is sharp for the multicolored sum-free problem: for
F2 by construction of Fu-Kleinberg, Fp by Kleinberg-Sawin-Speyer.
![Page 21: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/21.jpg)
Breakthrough
Theorem (Croot, Lev, Pach)
If A ⊂ Zn4 has no 3-AP, then |A| ≤ 4cn with c ≈ .926.
Theorem (Ellenberg, Gijswijt)
If A ⊂ Fnp has no 3-AP, then |A| ≤ p(1−cp)n for an explicit cp > 0.
Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, Alon
Same conclusion for the multicolored sum-free problem: If
{xi}mi=1,{yi}mi=1,{zi}mi=1 ⊂ Fnp with xi + yj + zk = 0⇔ i = j = k ,
then m ≤ p(1−cp)n.
Theorem
Exponent is sharp for the multicolored sum-free problem: for
F2 by construction of Fu-Kleinberg, Fp by Kleinberg-Sawin-Speyer.
![Page 22: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/22.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 23: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/23.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 24: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/24.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 25: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/25.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 26: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/26.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 27: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/27.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 28: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/28.jpg)
Proof sketch of the multicolored sum-free problem
Slice rank of tensors: Tao
A tensor T : [N]3 → F has slice rank 1 if there are functionsf : [N]→ F and g : [N]2 → F such that one of the following holds:
T (i , j , k) = f (i)g(j , k)
T (i , j , k) = f (i , k)g(j)
T (i , j , k) = f (i , j)g(k)
Slice rank of general tensor T : minimum number of rank onetensors needed to sum to T .
Claim
Diagonal tensor has rank equal to number of nonzero elements.
![Page 29: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/29.jpg)
Proof sketch of the multicolored sum-free problem
Let Mdn be the set of monomials of total degree at most d in n
variables, and degree less than p in each variable.
Claim
Take X = {x j}mj=1, Y = {y j}mj=1, Z = {z j}mj=1 in Fnp, as in the
multicolored sum-free problem. Then
m ≤ 3|M(p−1)n/3n |
Take a tensor T : (Fnp)3 → Fp:
T (x , y , z) =n∏
i=1
(1− (xi + yi + zi )p−1)
T is diagonal on X × Y × Z , so slice rank is at least m, and is at
most 3|M(p−1)n/3n |.
![Page 30: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/30.jpg)
Proof sketch of the multicolored sum-free problem
Let Mdn be the set of monomials of total degree at most d in n
variables, and degree less than p in each variable.
Claim
Take X = {x j}mj=1, Y = {y j}mj=1, Z = {z j}mj=1 in Fnp, as in the
multicolored sum-free problem. Then
m ≤ 3|M(p−1)n/3n |
Take a tensor T : (Fnp)3 → Fp:
T (x , y , z) =n∏
i=1
(1− (xi + yi + zi )p−1)
T is diagonal on X × Y × Z , so slice rank is at least m, and is at
most 3|M(p−1)n/3n |.
![Page 31: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/31.jpg)
Proof sketch of the multicolored sum-free problem
Let Mdn be the set of monomials of total degree at most d in n
variables, and degree less than p in each variable.
Claim
Take X = {x j}mj=1, Y = {y j}mj=1, Z = {z j}mj=1 in Fnp, as in the
multicolored sum-free problem. Then
m ≤ 3|M(p−1)n/3n |
Take a tensor T : (Fnp)3 → Fp:
T (x , y , z) =n∏
i=1
(1− (xi + yi + zi )p−1)
T is diagonal on X × Y × Z , so slice rank is at least m, and is at
most 3|M(p−1)n/3n |.
![Page 32: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/32.jpg)
Proof sketch of the multicolored sum-free problem
Let Mdn be the set of monomials of total degree at most d in n
variables, and degree less than p in each variable.
Claim
Take X = {x j}mj=1, Y = {y j}mj=1, Z = {z j}mj=1 in Fnp, as in the
multicolored sum-free problem. Then
m ≤ 3|M(p−1)n/3n |
Take a tensor T : (Fnp)3 → Fp:
T (x , y , z) =n∏
i=1
(1− (xi + yi + zi )p−1)
T is diagonal on X × Y × Z , so slice rank is at least m, and is at
most 3|M(p−1)n/3n |.
![Page 33: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/33.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every large graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tools in graph theory.
![Page 34: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/34.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every large graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tools in graph theory.
![Page 35: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/35.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every large graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tools in graph theory.
![Page 36: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/36.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every large graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tools in graph theory.
![Page 37: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/37.jpg)
Triangle Removal Lemma
Triangle Removal Lemma
For every ε > 0 there is δ > 0 such that if a n-vertex graph has at
most δn3 triangles, then we can delete at most εn2 edges and
remove all triangles.
Many applications in extremal graph theory, additive numbertheory, theoretical computer science, and combinatorics.
Proof uses Szemeredi’s regularity lemma, and gives a bound on1/δ which is a tower of two of height a power of 1/ε.
Problem (Alon, Erdos, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F.)
We may take 1/δ to be a tower of twos of height log 1/ε.
![Page 38: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/38.jpg)
Triangle Removal Lemma
Triangle Removal Lemma
For every ε > 0 there is δ > 0 such that if a n-vertex graph has at
most δn3 triangles, then we can delete at most εn2 edges and
remove all triangles.
Many applications in extremal graph theory, additive numbertheory, theoretical computer science, and combinatorics.
Proof uses Szemeredi’s regularity lemma, and gives a bound on1/δ which is a tower of two of height a power of 1/ε.
Problem (Alon, Erdos, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F.)
We may take 1/δ to be a tower of twos of height log 1/ε.
![Page 39: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/39.jpg)
Triangle Removal Lemma
Triangle Removal Lemma
For every ε > 0 there is δ > 0 such that if a n-vertex graph has at
most δn3 triangles, then we can delete at most εn2 edges and
remove all triangles.
Many applications in extremal graph theory, additive numbertheory, theoretical computer science, and combinatorics.
Proof uses Szemeredi’s regularity lemma, and gives a bound on1/δ which is a tower of two of height a power of 1/ε.
Problem (Alon, Erdos, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F.)
We may take 1/δ to be a tower of twos of height log 1/ε.
![Page 40: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/40.jpg)
Triangle Removal Lemma
Triangle Removal Lemma
For every ε > 0 there is δ > 0 such that if a n-vertex graph has at
most δn3 triangles, then we can delete at most εn2 edges and
remove all triangles.
Many applications in extremal graph theory, additive numbertheory, theoretical computer science, and combinatorics.
Proof uses Szemeredi’s regularity lemma, and gives a bound on1/δ which is a tower of two of height a power of 1/ε.
Problem (Alon, Erdos, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F.)
We may take 1/δ to be a tower of twos of height log 1/ε.
![Page 41: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/41.jpg)
Triangle Removal Lemma
Triangle Removal Lemma
For every ε > 0 there is δ > 0 such that if a n-vertex graph has at
most δn3 triangles, then we can delete at most εn2 edges and
remove all triangles.
Many applications in extremal graph theory, additive numbertheory, theoretical computer science, and combinatorics.
Proof uses Szemeredi’s regularity lemma, and gives a bound on1/δ which is a tower of two of height a power of 1/ε.
Problem (Alon, Erdos, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F.)
We may take 1/δ to be a tower of twos of height log 1/ε.
![Page 42: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/42.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 43: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/43.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 44: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/44.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 45: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/45.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 46: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/46.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 47: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/47.jpg)
Arithmetic Regularity Lemma
Let A ⊂ Fn3. The density of A in S is dA(S) = |A ∩ S |/|S |.
A translate S + x ⊂ Fn3 of a subspace S is ε-regular if
|dA(S + x)− dA(T )| ≤ ε
for every codimension 1 affine subspace T of S + x .
A subspace S is ε-regular if all but an ε-fraction of the translatesof S are ε-regular.
Green’s arithmetic regularity lemma
For each ε > 0 there is M(ε) such that for any A ⊂ Fn3 , there is
an ε-regular subspace S of codimension at most M(ε).
Green, Hosseini-Lovett-Moshkovitz-Shapira:M(ε) is a tower of twos of height ε−O(1).
![Page 48: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/48.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ⊂ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Green’s proof uses the arithmetic regularity lemma and gives abound on 1/δ which is a tower of two of height a power of 1/ε.
Kral’-Serra-Vena: new proof using graph triangle removal lemma.
Problem (Green)
Improve the bound in the arithmetic triangle removal lemma.
![Page 49: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/49.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ⊂ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Green’s proof uses the arithmetic regularity lemma and gives abound on 1/δ which is a tower of two of height a power of 1/ε.
Kral’-Serra-Vena: new proof using graph triangle removal lemma.
Problem (Green)
Improve the bound in the arithmetic triangle removal lemma.
![Page 50: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/50.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ⊂ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Green’s proof uses the arithmetic regularity lemma and gives abound on 1/δ which is a tower of two of height a power of 1/ε.
Kral’-Serra-Vena: new proof using graph triangle removal lemma.
Problem (Green)
Improve the bound in the arithmetic triangle removal lemma.
![Page 51: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/51.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ⊂ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Green’s proof uses the arithmetic regularity lemma and gives abound on 1/δ which is a tower of two of height a power of 1/ε.
Kral’-Serra-Vena: new proof using graph triangle removal lemma.
Problem (Green)
Improve the bound in the arithmetic triangle removal lemma.
![Page 52: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/52.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ⊂ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Green’s proof uses the arithmetic regularity lemma and gives abound on 1/δ which is a tower of two of height a power of 1/ε.
Kral’-Serra-Vena: new proof using graph triangle removal lemma.
Problem (Green)
Improve the bound in the arithmetic triangle removal lemma.
![Page 53: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/53.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 54: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/54.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 55: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/55.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 56: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/56.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 57: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/57.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 58: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/58.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 59: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/59.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X
v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 60: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/60.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 61: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/61.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 62: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/62.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 63: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/63.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 64: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/64.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 65: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/65.jpg)
Kral’, Serra, Vena proof
Fnp
Fnp
Fnp
u
v
w
u − v ∈ X v − w ∈ Y
w − u ∈ Z
Triangle x + y + z = 0corresponds to N := pn trianglesin the graph, and vice versa.
Thus, there are at most δN3
triangles.
Can remove εN2 edges and getrid of all triangles.
Remove x from X , Y , or Z if atleast N/3 edges corresponding toit are removed.
![Page 66: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/66.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239 andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 67: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/67.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239 andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 68: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/68.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239 andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 69: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/69.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239
andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 70: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/70.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239 andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 71: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/71.jpg)
Arithmetic Triangle Removal Lemma
A triangle in Fnp is a triple (x , y , z) of points with x + y + z = 0.
Green’s Arithmetic Triangle Removal Lemma
For every ε > 0 and prime p, there is δ > 0 such that if
X ,Y ,Z ∈ Fnp with at most δp2n triangles in X × Y × Z , then we
can delete εpn points and remove all triangles.
Much further work on bounds: Hatami-Sachdeva-Tulsiani,Bhattacharyya-Xie, Fu-Kleinberg, Haviv-Xie.
Theorem (F.-Lovasz)
We can take δ = (ε/3)Cp , where Cp = 1 + 1/cp is a computable
number. The exponent Cp is sharp.
In particular, C2 = 1 + 1/(5/3− log2 3) ≈ 13.239 andC3 = 1 + 1/c3 where c3 = 1− log b
log 3 , b = a−2/3 + a1/3 + a4/3, and
a =√
33−18 , so C3 ≈ 13.901.
![Page 72: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/72.jpg)
Removal lemma proof sketch
Theorem (F., L. M. Lovasz)
With δ = (ε/3)Cp , if X ,Y ,Z ⊂ Fnp have at most δp2n triangles in
X ×Y ×Z , then we can delete εpn points and remove all triangles.
Goal 1
With δ = εCp , the union of any εN disjoint triangles with elements
red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 2
With δ = εCp+o(1), the union of any εN disjoint triangles with
elements red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
![Page 73: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/73.jpg)
Removal lemma proof sketch
Theorem (F., L. M. Lovasz)
With δ = (ε/3)Cp , if X ,Y ,Z ⊂ Fnp have at most δp2n triangles in
X ×Y ×Z , then we can delete εpn points and remove all triangles.
Goal 1
With δ = εCp , the union of any εN disjoint triangles with elements
red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 2
With δ = εCp+o(1), the union of any εN disjoint triangles with
elements red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
![Page 74: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/74.jpg)
Removal lemma proof sketch
Theorem (F., L. M. Lovasz)
With δ = (ε/3)Cp , if X ,Y ,Z ⊂ Fnp have at most δp2n triangles in
X ×Y ×Z , then we can delete εpn points and remove all triangles.
Goal 1
With δ = εCp , the union of any εN disjoint triangles with elements
red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 2
With δ = εCp+o(1), the union of any εN disjoint triangles with
elements red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
![Page 75: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/75.jpg)
Removal lemma proof sketch
Theorem (F., L. M. Lovasz)
With δ = (ε/3)Cp , if X ,Y ,Z ⊂ Fnp have at most δp2n triangles in
X ×Y ×Z , then we can delete εpn points and remove all triangles.
Goal 1
With δ = εCp , the union of any εN disjoint triangles with elements
red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 2
With δ = εCp+o(1), the union of any εN disjoint triangles with
elements red, yellow, blue have ≥ δN2 rainbow triangles.
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
![Page 76: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/76.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 77: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/77.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 78: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/78.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 79: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/79.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 80: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/80.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 81: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/81.jpg)
Arithmetic triangle removal lemma proof idea
Goal 3
With δ = εCp+o(1), if we have εN disjoint rainbow triangles with
each element in ≈ βN rainbow triangles, then β ≥ δ/ε.
Sample a random affine subspace S with |S | ≈ 1/β elements.
A rainbow triangle is good if each of its elements are in exactly onerainbow triangle in S .
With positive probability, the densities of X ,Y ,Z in S are ≈ ε anda constant fraction of the elements are in good rainbow triangles.
From the multicolor sum-free theorem
ε� |S |−cp ≈ (1/β)−cp ,
which gives δ ≤ εCp+o(1).
![Page 82: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/82.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Quiz
How large is n(ε)?
(a) Θ(log(1/ε))
(b) ε−Θ(1)
(c) 2ε−Θ(1)
(d) Tower (Θ (log(1/ε)))
(e) Tower(Θ((1/ε)Θ(1)
))
![Page 83: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/83.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Quiz
How large is n(ε)?
(a) Θ(log(1/ε))
(b) ε−Θ(1)
(c) 2ε−Θ(1)
(d) Tower (Θ (log(1/ε)))
(e) Tower(Θ((1/ε)Θ(1)
))
![Page 84: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/84.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Quiz
How large is n(ε)?
(a) Θ(log(1/ε))
(b) ε−Θ(1)
(c) 2ε−Θ(1)
(d) Tower (Θ (log(1/ε)))
(e) Tower(Θ((1/ε)Θ(1)
))
![Page 85: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/85.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Quiz
How large is n(ε)?
(a) Θ(log(1/ε))
(b) ε−Θ(1)
(c) 2ε−Θ(1)
(d) Tower (Θ (log(1/ε)))
(e) Tower(Θ((1/ε)Θ(1)
))
![Page 86: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/86.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Quiz
How large is n(ε)?
(a) Θ(log(1/ε))
(b) ε−Θ(1)
(c) 2ε−Θ(1)
(d) Tower (Θ (log(1/ε)))
(e) Tower(Θ((1/ε)Θ(1)
))
![Page 87: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/87.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Theorem (F.-Pham)
n(ε) = Tower (Θ (log(1/ε)))
This is the first application of a regularity lemma where atower-type bound is shown to be needed.
Theorem* (F.-Pham-Zhao)
A similar result holds in abelian groups and in [N].
![Page 88: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/88.jpg)
Progressions with popular differences
Theorem (Green)
∀ ε > 0 there is a least n(ε) such that if n ≥ n(ε), then ∀ A ⊂ Fn3
of density α, there is a nonzero d such that the density of 3-term
arithmetic progressions with common difference d is at least α3− ε.
Theorem (F.-Pham)
n(ε) = Tower (Θ (log(1/ε)))
This is the first application of a regularity lemma where atower-type bound is shown to be needed.
Theorem* (F.-Pham-Zhao)
A similar result holds in abelian groups and in [N].
![Page 89: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/89.jpg)
Half the random bound
Definition
Let n′(α) be the least integer such that if n ≥ n′(α), then for every
A ⊂ Fn29 of density α, there is a nonzero d such that the density of
3-term APs with common difference d is at least α3/2.
Quiz
How large is n′(α)?
(a) Θ(log(1/α))
(b) α−Θ(1)
(c) 2α−Θ(1)
(d) Tower (Θ (log log(1/α)))
(e) Tower (Θ (log(1/α)))
![Page 90: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/90.jpg)
Half the random bound
Definition
Let n′(α) be the least integer such that if n ≥ n′(α), then for every
A ⊂ Fn29 of density α, there is a nonzero d such that the density of
3-term APs with common difference d is at least α3/2.
Quiz
How large is n′(α)?
(a) Θ(log(1/α))
(b) α−Θ(1)
(c) 2α−Θ(1)
(d) Tower (Θ (log log(1/α)))
(e) Tower (Θ (log(1/α)))
![Page 91: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/91.jpg)
Half the random bound
Definition
Let n′(α) be the least integer such that if n ≥ n′(α), then for every
A ⊂ Fn29 of density α, there is a nonzero d such that the density of
3-term APs with common difference d is at least α3/2.
Quiz
How large is n′(α)?
(a) Θ(log(1/α))
(b) α−Θ(1)
(c) 2α−Θ(1)
(d) Tower (Θ (log log(1/α)))
(e) Tower (Θ (log(1/α)))
![Page 92: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/92.jpg)
Half the random bound
Definition
Let n′(α) be the least integer such that if n ≥ n′(α), then for every
A ⊂ Fn29 of density α, there is a nonzero d such that the density of
3-term APs with common difference d is at least α3/2.
Quiz
How large is n′(α)?
(a) Θ(log(1/α))
(b) α−Θ(1)
(c) 2α−Θ(1)
(d) Tower (Θ (log log(1/α)))
(e) Tower (Θ (log(1/α)))
![Page 93: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/93.jpg)
Half the random bound
Definition
Let n′(α) be the least integer such that if n ≥ n′(α), then for every
A ⊂ Fn29 of density α, there is a nonzero d such that the density of
3-term APs with common difference d is at least α3/2.
Theorem (F.-Pham)
n′(α) = Tower (Θ (log log(1/α)))
![Page 94: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/94.jpg)
Multidimensional cap set problem
Inspired by an idea of David Fox on the game SET.
Definition
Let r(n, d) be the maximum |A| over all A ⊂ Fn3 which contains no
d-dimensional afffine subspace.
Theorem:
For N = 3n, we have N1−(d+1)3−d ≤ r(n, d) ≤ N1−13.902−d.
Hence, the largest dimension of an affine subspace guaranteed in
any subset of Fn3 of density α is Θ
(log(
nlog(1/α)
)).
Question:
Asymptotics?
![Page 95: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/95.jpg)
Multidimensional cap set problem
Inspired by an idea of David Fox on the game SET.
Definition
Let r(n, d) be the maximum |A| over all A ⊂ Fn3 which contains no
d-dimensional afffine subspace.
Theorem:
For N = 3n, we have N1−(d+1)3−d ≤ r(n, d) ≤ N1−13.902−d.
Hence, the largest dimension of an affine subspace guaranteed in
any subset of Fn3 of density α is Θ
(log(
nlog(1/α)
)).
Question:
Asymptotics?
![Page 96: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/96.jpg)
Multidimensional cap set problem
Inspired by an idea of David Fox on the game SET.
Definition
Let r(n, d) be the maximum |A| over all A ⊂ Fn3 which contains no
d-dimensional afffine subspace.
Theorem:
For N = 3n, we have N1−(d+1)3−d ≤ r(n, d) ≤ N1−13.902−d.
Hence, the largest dimension of an affine subspace guaranteed in
any subset of Fn3 of density α is Θ
(log(
nlog(1/α)
)).
Question:
Asymptotics?
![Page 97: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/97.jpg)
Multidimensional cap set problem
Inspired by an idea of David Fox on the game SET.
Definition
Let r(n, d) be the maximum |A| over all A ⊂ Fn3 which contains no
d-dimensional afffine subspace.
Theorem:
For N = 3n, we have N1−(d+1)3−d ≤ r(n, d) ≤ N1−13.902−d.
Hence, the largest dimension of an affine subspace guaranteed in
any subset of Fn3 of density α is Θ
(log(
nlog(1/α)
)).
Question:
Asymptotics?
![Page 98: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/98.jpg)
Multidimensional cap set problem
Inspired by an idea of David Fox on the game SET.
Definition
Let r(n, d) be the maximum |A| over all A ⊂ Fn3 which contains no
d-dimensional afffine subspace.
Theorem:
For N = 3n, we have N1−(d+1)3−d ≤ r(n, d) ≤ N1−13.902−d.
Hence, the largest dimension of an affine subspace guaranteed in
any subset of Fn3 of density α is Θ
(log(
nlog(1/α)
)).
Question:
Asymptotics?
![Page 99: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/99.jpg)
Open Problems
Better estimate the bound on the cap set problem.
Prove good estimates for the Green-Tao analogue of Green’spopular difference theorem for 4-term APs.
Obtain reasonable bounds on Roth’s theorem and the arithmetictriangle removal lemma in other abelian groups.
Better estimate the bounds on higher dimensional cap sets.
Extend the new cap set theorem to longer arithmetic progressions.
![Page 100: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/100.jpg)
Open Problems
Better estimate the bound on the cap set problem.
Prove good estimates for the Green-Tao analogue of Green’spopular difference theorem for 4-term APs.
Obtain reasonable bounds on Roth’s theorem and the arithmetictriangle removal lemma in other abelian groups.
Better estimate the bounds on higher dimensional cap sets.
Extend the new cap set theorem to longer arithmetic progressions.
![Page 101: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/101.jpg)
Open Problems
Better estimate the bound on the cap set problem.
Prove good estimates for the Green-Tao analogue of Green’spopular difference theorem for 4-term APs.
Obtain reasonable bounds on Roth’s theorem and the arithmetictriangle removal lemma in other abelian groups.
Better estimate the bounds on higher dimensional cap sets.
Extend the new cap set theorem to longer arithmetic progressions.
![Page 102: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/102.jpg)
Open Problems
Better estimate the bound on the cap set problem.
Prove good estimates for the Green-Tao analogue of Green’spopular difference theorem for 4-term APs.
Obtain reasonable bounds on Roth’s theorem and the arithmetictriangle removal lemma in other abelian groups.
Better estimate the bounds on higher dimensional cap sets.
Extend the new cap set theorem to longer arithmetic progressions.
![Page 103: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/103.jpg)
Open Problems
Better estimate the bound on the cap set problem.
Prove good estimates for the Green-Tao analogue of Green’spopular difference theorem for 4-term APs.
Obtain reasonable bounds on Roth’s theorem and the arithmetictriangle removal lemma in other abelian groups.
Better estimate the bounds on higher dimensional cap sets.
Extend the new cap set theorem to longer arithmetic progressions.
![Page 104: Jacob Fox Stanford University Matematick e Kolokvium 99 ...iuuk.mff.cuni.cz/events/doccourse/FOX.pdf · Arithmetic regularity, removal, and progressions Jacob Fox Stanford University](https://reader031.fdocuments.net/reader031/viewer/2022041503/5e2347cdf1f92e25fb1c3585/html5/thumbnails/104.jpg)
Thank you!