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A variant of the Corners theorem

Part of: Sequences and sets Extremal combinatorics

Published online by Cambridge University Press:  02 February 2021

MATEI MANDACHE
MATEI MANDACHE*
Affiliation:
Mathematical Institute, University of Oxford, Oxford. e-mail: matei.mandache@gmail.com
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Abstract

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| ≥ αN2 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 Sd={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |Sd| 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 |Sd|>(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 |Sd|>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 |Sd|>(α4-ϵ)N2.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 05D05: Extremal set theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 607 - 621
Copyright
© Cambridge Philosophical Society 2021

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