Bohr-set density increment lemma
= Bohr-set density increment lemma
{c}
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 either
$$
|A|\leq(d/\alpha)^{O(d)}|B|^{1/2},
$$
or some translate of $A$ has relative density at least $(1+c\alpha)\alpha$ in a regular <Bohr set> $B'\subseteq B$ of rank at most $d+1$ and width at least $\rho(B)(\alpha/d)^{O(1)}$.