Roth's theorem

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Roth's theorem ([math]{\Bbb Z}/N{\Bbb Z}[/math] version) If N is sufficiently large depending on [math]\delta \gt 0[/math], then any subset A of [math][N][/math] of density at least [math]\delta[/math] contains an arithmetic progression x, x+r, x+2r with [math]r \gt 0[/math].

Roth-Meshulam theorem ([math][3]^n[/math] version) If n is sufficiently large depending on [math]\delta \gt 0[/math], then any subset of [math][3]^n[/math] of density at least [math]\delta[/math] contains an algebraic line, i.e. a triple (x,x+r,x+2r) where r is non-zero and we identify [math][3]^n[/math] with [math]({\Bbb Z}/3{\Bbb Z})^n[/math].

Roth's theorem is a special case of Szemerédi's theorem. See also Szemerédi's combinatorial proof of Roth's theorem.

Roth's theorem is implied by the corners theorem, which in turn is implied by the k=3 case of the IP-Szemerédi theorem, which is in turn implied by k=3 case of the density Hales-Jewett theorem.