# Roth's theorem

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