$2c^{-1} + \log_2(\delta^{-1})$. Since this cannot happen, it follows that there exists $n {\leqslant}\log_2(\delta^{-1})+O(1)$ such that the procedure terminates at stage $n$.

At the point of termination, the smallness assumption must hold, so that $$\label{N_i upper bound}
N_n {\leqslant}\exp{\left( O{\Bigl( \delta^{-O(1)} \Bigr)} \right)}.$$ On the other hand, iteratively applying the lower bound , we have $$\label{N_i lower bound}
\begin{split}
N_n  & {\geqslant}\frac{N_{n-1}^{1/2}}{{\left( q_0^{2^{n-1}}\dotsm q_{n-1}\exp{\left( \delta^{-O(1)} \right)} \right)}^{O(1)}}\\
&  {\geqslant}N^{1/2^n}{\left[ q_0^{2^{n-1}}\dotsm q_{n-1}\exp{\left( \delta^{-O(1)} \right)} \right]}^{-O(1 + {\frac{1}{2}} + {\frac{1}{4}} + \dots + 2^{1-n}) } \\
& \gg \exp{\left( -O{\left( \delta^{-O(1)} \right)} \right)} N^{\Omega(\delta)},
\end{split}$$ where we use the upper bound on the $q_i$’s, together with $n {\leqslant}\log_2(1/\delta)+O(1)$. Taking a logarithm and comparing upper and lower bounds for $N_n$ gives $
\log N \ll \delta^{-O(1)},
$ which yields the bound claimed in Theorem \[main\].

The cut norm inverse theorem {#inverse theorem section}
============================

The first step of the proof of Lemma \[increment lemma\] is to use the main technical result of [@PelusePrendivilleQuantitative] to prove an inverse theorem for the cut norm associated to $\Lambda_{q,N}$, which we now define.

For positive integers $q {\leqslant}N$, we define the *cut norm* of $f : {\mathbb{Z}}\to {\mathbb{C}}$ by $$\label{q norm eq}
{\left\| f\right\|}_{q, N} := \sup\{|\Lambda_{q, 