that for each $i$, $$E(t_i \cdot S_i, S_{i+1}) {\geqslant}\frac{{\varepsilon}^2}{576 K_1^4}|S|^3.$$

Now, let us apply Lemma \[lem:gs\_doubling\_lem\] to $S_i {\text{ and }}S_{i+1}$. We have that $$E(t_i \cdot S_i, S_{i+1}) {\geqslant}\frac{{\varepsilon}^2}{576 K_1^4} |S_i|^{3/2}|S_{i+1}|^{3/2},$$ and so we deduce that $$\begin{aligned}
|t_i \cdot S_i - S_{i+1}| &{\leqslant}\delta[S_i] \delta[S_{i+1}] |S_i|^{1/2}|S_{i+1}|^{1/2} \frac{576K_1^4}{{\varepsilon}^2} \nonumber\\
&{\leqslant}\frac{576 K_1^4K_2^2}{{\varepsilon}^2}|S_i|^{1/2}|S_{i+1}|^{1/2}.\end{aligned}$$

Now, we can prove, by inductively applying Lemma \[lem:ruzsa\_triangle\], that $$|t_1t_2 \dots t_i \cdot S_1 - S_{i+1}| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^i |S_1|^{1/2}|S_{i+1}|^{1/2}.$$ Thus, setting $i=k$, we learn that $$|t_1t_2 \dots t_k \cdot S_1 - S_1| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1} |S_1|,$$ since $k {\leqslant}K_1$.

By hypothesis, $t_1t_2 \dots t_k \neq 1$. However, we know that $t_1t_2\dots t_k$ can be written in the form $M^{e_1}(M+1)^{e_2}$ for some integers $e_i$ not both zero. Suppose that $e_1$ is nonzero; the argument is similar if $e_2$ is nonzero.

Write $t_1t_2 \dots t_k = \frac{r}{s}$ for coprime integers $r {\text{ and }}s$; our hypothesis tells us that $M$ must divide $r$ or $s$. Therefore, $$|r| + |s| > \left(\frac{1000 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1},$$ as a consequence of (\[eqn:condition\_on\_M\]).

Thus, we have shown that $|r \cdot S_1 - s \cdot S_1| {\leqslant}\left(\frac{576 K_1^4K_2^2}{{\varepsilon}^2}\right)^{K_1} |S_1|$. But, if $S_1$ is sufficiently large, this contradicts Theorem \[thm:bukh\_sod\_thm\], which states that $$|r \cdot S_1 - s \cdot S_1| > 