Proof of a conjecture of Kleinberg-Sawin-Speyer

Pebody, Luke

doi:10.19086/da.3733

Cited by 16 publications

(27 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the next subsection, we will introduce scaled distributions, following Pebody [, Section 2]. They provide a useful framework for the proof of Theorem .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Here, we will introduce scaled distributions, the framework in which the proof of Theorem will operate. Everything in this subsection follows the first half of Section 2 of Pebody's paper . Throughout this subsection,

n

is an arbitrary nonnegative integer.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“… established the upper bound, Kleinberg, Sawin, and Speyer proved Theorem for

k = 3

and any

m ⩾ 2

. Their proof uses a statement that had been formulated as a conjecture in an earlier version of their paper and was then proved by Pebody . In order to make the statement precise, we need some more notation.…”

Section: Introductionmentioning

confidence: 99%

“…Our proof of Theorem was inspired by the first version of Pebody's proof of the conjecture of Kleinberg, Sawin, and Speyer. However, our proof is not a direct generalization of Pebody's work and if one restricts to

k = 3

in our proof, one obtains a significantly different proof.…”

Section: Introductionmentioning

confidence: 99%

“…However, our proof is not a direct generalization of Pebody's work and if one restricts to

k = 3

in our proof, one obtains a significantly different proof. In particular, we avoid the large case analysis in , which would become even larger when trying to generalize it to

k > 3

. Pebody later replaced the first version of his proof by yet another, much shorter proof .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A lower bound for the k‐multicolored sum‐free problem in Zmn

Lovász

Sauermann

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

In this paper, we give a lower bound for the maximum size of a k‐colored sum‐free set in Zmn, where k⩾3 and m⩾2 are fixed and n tends to infinity. If m is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a k‐colored sum‐free set in Zmn. This generalizes a result of Kleinberg–Sawin–Speyer for the case k=3 and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg–Sawin–Speyer. Both of these generalizations require several key new ideas.

show abstract

“…In the next subsection, we will introduce scaled distributions, following Pebody [, Section 2]. They provide a useful framework for the proof of Theorem .…”

Section: Proof Of Theoremmentioning

confidence: 99%

n

is an arbitrary nonnegative integer.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“… established the upper bound, Kleinberg, Sawin, and Speyer proved Theorem for

k = 3

and any

m ⩾ 2

Section: Introductionmentioning

confidence: 99%

k = 3

in our proof, one obtains a significantly different proof.…”

Section: Introductionmentioning

confidence: 99%

“…However, our proof is not a direct generalization of Pebody's work and if one restricts to

k = 3

in our proof, one obtains a significantly different proof. In particular, we avoid the large case analysis in , which would become even larger when trying to generalize it to

k > 3

. Pebody later replaced the first version of his proof by yet another, much shorter proof .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A lower bound for the k‐multicolored sum‐free problem in Zmn

Lovász

Sauermann

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

Upper Bounds For Families Without Weak Delta-Systems

Naslund

2023

Combinatorica

View full text Add to dashboard Cite

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

Fox

Lovász

2017

Advances in Mathematics

View full text Add to dashboard Cite

Let p be a fixed prime. A triangle in F n p is an ordered triple (x, y, z) of points satisfying x + y + z = 0. Let N = p n = |F n p |. Green proved an arithmetic triangle removal lemma which says that for every ǫ > 0 and prime p, there is a δ > 0 such that if X, Y, Z ⊂ F n p and the number of triangles in X × Y × Z is at most δN 2 , then we can delete ǫN elements from X, Y , and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/δ can be taken to be an exponential tower of twos of height logarithmic in 1/ǫ.We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F n p . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a explicit number C p such that we may take δ = (ǫ/3) Cp , and we must have δ ≤ ǫ Cp−o(1) . In particular, C 2 = 1 + 1/(5/3 − log 2 3) ≈ 13.239, and C 3 = 1 + 1/c 3 with c 3 = 1 − log b log 3 , b = a −2/3 + a 1/3 + a 4/3 , and a = √ 33−1 8, which gives C 3 ≈ 13.901. The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg-SawinSpeyer, Norin, and Pebody, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-GrochowNaslund-Sawin-Umans, and Alon.

show abstract

Proof of a conjecture of Kleinberg-Sawin-Speyer

Cited by 16 publications

References 5 publications

A lower bound for the k‐multicolored sum‐free problem in Zmn

A lower bound for the k‐multicolored sum‐free problem in Zmn

Upper Bounds For Families Without Weak Delta-Systems

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

Contact Info

Product

Resources

About