I would like to understand here on page $11$ in the proof of LEMMA $3.2.$ why
sinc $k<b/3$ the only block of $b$ consecutive $3$'s are $b_{q+1}$ and (perhaps) $3^s$
I.e. how $k<b/3$ has been used there ? I'm skipping to $11$ page of a (moderately) technical paper since I'm reading it carefully as it is interesting with the hope that someone else find it interesting too to read it up to that point.
What is really being used, so far as I can see, is that $k<b$. We know that $yB_pC_pB_{p+1}z$ contains two blocks of $b$ consecutive $3$s, one in $B_p$ and one in $B_{p+1}$, so $3^rC_qB_{q+1}C_{q+1}3^sw$ must also contain two such blocks. $B_{q+1}$ contains one of them. From the top of page $10$ we know that each of $C_q$ and $C_{q+1}$ contains exactly $k$ $3$s, and $k<b$, so neither $C_q$ nor $C_{q+1}$ contains a block of $b$ consecutive $3$s. Thus, the only possible locations for the other block are $3^r$ and $3^s$. However, clause (i) of the definition of type $2$ sequences requires that $r<b$, so $3^s$ is actually the only possible place of the other block of $b$ consecutive $3$s, and therefore we must have $s=b$, matching $B_{p+1}$. This implies that $B_p=B_{q+1}$ and hence that $p=q+1$, and also that $y=3^rC_q=3^rC_{p-1}$. Finally, $y$ is a tail of $C_{p-1}$, so we must have $r=0$, but then $y=C_{p-1}$, contradicting the requirement in the definition of type $1$ subsequence that it be a proper tail of $C_{p-1}$.