Bourgain bound for three-term-progression-free sets

If $G$ is a finite abelian group of odd order $N$ and $A\subseteq G$ contains no nonconstant three-term arithmetic progression, then Bourgain's bound is

Let $A$ have relative density $\alpha$ in a rank-$d$ Regular Bohr set $B$, and suppose $A$ has no nonconstant three-term arithmetic progression. Then eitheror some translate of $A$ has relative density at least $(1+c\alpha )\alpha$ in a regular Bohr set $B^{\prime }\subseteq B$ of rank at most $d+1$ and width at least $\rho (B)(\alpha /d{)}^{O(1)}$.