Tower-type bounds for Roth’s theorem with popular differences

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on progressions in dense sets. It states that for every $\epsilon > 0$ there is some $N\_0(\epsilon)$ such $N \ge N\_0(\epsilon)$ and $A \subset \[N]$ with $|A| = \alpha N$, nonzero $d$ $A$ contains at least $(\alpha^3 - \epsilon) N$ three-term common difference $d$. We the minimum Green's exponential tower twos height order $\log(1/\epsilon)$. Both lower upper bounds are new. This shows tower-type arise from use this application quantitatively necessary.