us suppose we have a sequence of tuples $(A_i, N_i, q_i)$ for $i = 0, 1, \dots, n$ that each satisfy the following:

(i) $A_i$ lacks configurations of the form $$x,\ x+y,\ x+q_0^{2^i} q_1^{2^{i-1}}\dotsm q_{i-1}^2 q_i y^2 \qquad (y \neq 0).$$

(ii) \[qi upper bound\] $q_i {\leqslant}\exp{\left( O{\left( \delta^{-O(1)} \right)} \right)}$;

(iii) $A_i \subset [N_i]$ and for $i {\geqslant}1$ we have $$\frac{|A_i|}{N_i} {\geqslant}(1+c)\frac{|A_{i-1}|}{N_{i-1}},$$ where $c = \Omega(1)$ is a positive absolute constant;

(iv) \[length lower bound\] for $i {\geqslant}1$ we have the lower bound $$N_i {\geqslant}\frac{N_{i-1}^{1/2}}{{\left( q_0^{2^{i-1}}\dotsm q_{i-1}\exp{\left( \delta^{-O(1)} \right)} \right)}^{O(1)}}.$$

By Lemma \[increment lemma\], either $$\label{termination condition}
N_n \ll {\left( q_0^{2^n}q_1^{2^{n-1}}\dotsm q_{n-1}^2 q_n/\delta \right)}^{O(1)},$$ or we may obtain $(A_{n+1}, N_{n+1}, q_{n+1})$ satisfying conditions (i)–(iv). If holds, then our iterative process terminates at stage $n$.

If the number of iterations $n$ is at least $c^{-1}$, then the density of $A_n$ on $[N_n]$ is at least $2\delta$. After an additional ${\tfrac{1}{2}}c^{-1}$ iterations, the density is at least $4\delta$. Hence if the number of iterations is at least $${\left\lceil c^{-1} \right\rceil} + {\left\lceil {\tfrac{1}{2}}c^{-1} \right\rceil} + {\left\lceil {\tfrac{1}{4}}c^{-1} \right\rceil}+ \dots + {\left\lceil {\tfrac{1}{2^{m-1}}}c^{-1} \right\rceil},$$ then the density is at least $2^m\delta$. The density therefore exceeds one if the number of iterations exceeds 