Lemma 2.24. Assuming that:

$A \subseteq [N]$ of density $α > 0$
$p$ a prime in $[\frac{N}{3}, \frac{2 N}{3}]$
let $A^{'} = A \cap [p] \subseteq ℤ ∕ p ℤ$
$| \hat{𝟙_{A^{'}}} (t) | \geq \frac{α^{2}}{20}$ for some $t \neq 0$

Then there exists a progression

P \subseteq [N]

of length at least

α^{2} \frac{\sqrt{N}}{500}

such that

| A \cap P | \geq α (1 + \frac{α}{80}) | P |

.