\section{The counting consequence of the earlier bound}
\label{app:earlier-count}

We justify the bound $I(s)\ge0.21\sqrt{s/\log s}$ used in the
introduction to compare the present result with \cite{Tind}.
The application of Fischler's criterion in \cite[Section~7]{Tind}
gives a lower bound for the span including $1$. With the parameter
estimates from that section, it yields
\[
 \dim_\Q\operatorname{span}_\Q\{1,\beta(2),\ldots,\beta(s)\}
 \ge 1+(c_*+o(1))\sqrt{\frac{s}{\log s}},
 \qquad c_*=0.2174562348\ldots,
\]
as $s$ tends to infinity through multiples of $34$.
All rational beta values lie in $\Q\cdot1$, and each irrational
value adds at most one dimension. The dimension on the left is
therefore at most $1+I(s)$, so
\[
 I(s)\ge(c_*+o(1))\sqrt{\frac{s}{\log s}}
 \qquad(34\mid s).
\]

For an arbitrary large even $s$, put $t=34\lfloor s/34\rfloor$.
Then $I(s)\ge I(t)$ and
\[
 \frac{\sqrt{t/\log t}}{\sqrt{s/\log s}}\longrightarrow1.
\]
Since $c_*>0.21$, the preceding estimate gives
$I(s)\ge0.21\sqrt{s/\log s}$ for every sufficiently large even $s$.