Corners theorem

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Corners theorem: ( ${\Bbb Z}/N{\Bbb Z}$ version) If N is sufficiently large depending on $δ$ , then any $δ$ -dense subset of $[N] 2$ must contain a "corner" (x,y), (x+r,y), (x,y+r) with $r > 0$ .

Corners theorem: ( $({\Bbb Z}/3{\Bbb Z})^n$ version) If n is sufficiently large depending on $δ$ , then any $δ$ -dense subset of ${}(({\Bbb Z}/3{\Bbb Z})^n)^2$ must contain a "corner" (x,y), (x+r,y), (x,y+r) with $r \neq 0$ .

This result was first proven by Ajtai and Szemerédi. A simpler proof, based on the triangle removal lemma, was obtained by Solymosi. The corners theorem implies Roth's theorem and is in turn implied by the IP-Szemerédi theorem, which in turn follows from DHJ(3).

One consequence of the corners theorem is that any subset of the triangular grid

$\Delta_n = \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n\}$

of density at least $δ$ will contain an equilateral triangle $(a + r, b, c),(a, b + r, c),(a, b, c + r)$ with $r > 0$ , if n is sufficiently large depending on $δ$ .

A special case of the corners theorem is of interest in connection with DHJ(1,3).

Corners theorem

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