Triforce and corners

Fox, Jacob; Sah, Ashwin; Sawhney, Mehtaab; Stoner, David; Zhao, Yufei

doi:10.1017/s0305004119000173

Cited by 12 publications

(21 citation statements)

References 17 publications

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“…. Meanwhile, since α 4 is a convex function of α, we can deduce that for the family G n = F n 2 , M(α) ≥ α 4 . This last bound also holds for the family G n = F n p for any prime p. It would be natural to ask what is the correct exponent log(m(α))/ log(α)).…”

Section: •3 Conclusionmentioning

confidence: 95%

“…we refine our partitions by replacing B i with B and B i \ B and replacing C j with C and C j \ C , we can apply the second part of Lemma 2 using 4 3 .…”

Section: •2 Refinement (B)mentioning

confidence: 99%

“…Since this paper was made public on arxiv.org [9], there have been further developments. In [4], Fox et al show that m(α) is α 4−o (1) , but not O(α 4 ), as α → 0.…”

Section: Introductionmentioning

confidence: 99%

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A variant of the Corners theorem

Mandache¹

2021

Math. Proc. Camb. Phil. Soc.

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The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN 2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A. Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define S d ={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |S d | is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |S d |>(α3-ϵ)N2? We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets A with |S d |>Cα3.13N2 for all d ≠ 0, where C is an absolute constant. We also show that in the special case where $G = {\mathbb{F}}_2^n$ , one can always find a d with |S d |>(α4-ϵ)N2.

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Section: •3 Conclusionmentioning

confidence: 95%

“…we refine our partitions by replacing B i with B and B i \ B and replacing C j with C and C j \ C , we can apply the second part of Lemma 2 using 4 3 .…”

Section: •2 Refinement (B)mentioning

confidence: 99%

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A variant of the Corners theorem

Mandache¹

2021

Math. Proc. Camb. Phil. Soc.

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“…Fox, Sah, Sawhney, Stoner and Zhao [4] determined more precise asymptotics, showing:…”

Section: M(α) ≤ M(α)mentioning

confidence: 99%

“…In a somewhat surprising difference from the k-AP case, Mandache shows that the exponents in the growth rates of m(α), m (α) are strictly greater than 3, whereas random subsets of G × G have approximately α 3 |G| 2 corners for each fixed difference d = 0. The asymptotic growth rates of m and m were recently determined by Fox, Sah, Sawhney, Stoner, and Zhao [4], who also discuss other possible generalisations and barriers to generalisation for popular differences results. We will include their bounds on m and m following the discussion of the variational problem itself.…”

Section: Introductionmentioning

confidence: 99%

Popular differences for corners in Abelian groups

Berger¹

2020

Math. Proc. Camb. Phil. Soc.

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For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G. Recently, Mandache proved a “popular differences” version of this result in the finite field case $G = {\mathbb{F}}_p^n$ , showing that for any subset A ⊂ G × G of density α, one can fix d ≠ 0 such that A contains a large fraction, now known to be approximately α4, of all corners with difference d, as x, y vary over G. We generalise Mandache’s result to all compact abelian groups G, as well as the case of corners in $\mathbb{Z}^2$ .

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Khintchine-type recurrence for 3-point configurations

Ackelsberg

Bergelson

2022

Forum of Mathematics, Sigma

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The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi , \psi \}$ has the large intersections property. That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10).

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Triforce and corners

Abstract: AbstractMay the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).Let M(δ Show more

Cited by 12 publications

References 17 publications

A variant of the Corners theorem

A variant of the Corners theorem

Popular differences for corners in Abelian groups

Khintchine-type recurrence for 3-point configurations

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