Proof of Theorem 1.1 — $\nu_\ell=\lfloor c\,\ell\ln(2/\varepsilon)\rfloor$ gives $\beta\le\varepsilon$ at cost $O(k\ln(2/\varepsilon))$

There are absolute constants $c>0$ and $C>0$ such that the following holds for every $\theta\ge1$ and every $\varepsilon\in(0,1]$. Set

Then

1. every $\nu_\ell$ is a positive integer;
2. $\displaystyle\beta(\nu_1,\dots,\nu_\theta)=\prod_{\ell=1}^{\theta}\frac{1}{\cos\big(\frac{\pi}{2^{\nu_\ell+1}}\big)}-1\le\varepsilon;$
3. $\displaystyle\sum_{\ell=1}^{\theta}2^{\theta-\ell}\nu_\ell\le C\,2^\theta\ln\frac{2}{\varepsilon}.$

With $k=2^\theta$, item 3 is the estimate that turns the size bounds $p\le k+O(1)\sum_\ell 2^{\theta-\ell}\nu_\ell$ and $q\le O(1)\sum_\ell 2^{\theta-\ell}\nu_\ell$ of the approximation (10) into $p,q\le O(1)\,k\ln(2/\varepsilon)$, as claimed in Theorem 1.1.

Formalization Note The paper writes $\nu_\ell=\lfloor O(1)\,\ell\ln(2/\varepsilon)\rfloor$ "with properly chosen absolute constant $O(1)$"; here that constant is the existential $c$, chosen before $\theta$ and $\varepsilon$, and $C$ likewise. Item 1 is explicit because the construction (8) needs positive integers. The paper's conclusion is stated for $p$ and $q$; since those depend on how (10) is encoded as a linear map, this statement records the arithmetic estimate on $\sum_\ell 2^{\theta-\ell}\nu_\ell$ that the paper's size bounds reduce to.