# Roth's theorem

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