# Branching Densities of Cube-Free and Square-Free Words
## 3 Positions Fixed by Big Cubes
Then the mutual location of the suffix \(QQQ\) and factor \(PPP^{\prime}\) of \(w\) is as in Figure 6 (\(x\
eq\lambda\) since \(2q>p\); \(v\
eq\lambda\) since \(q<p\); \(z\
eq\lambda\); since \(2q<l+p\); \(y\) can be empty). Since \(x,vx\) are prefixes of \(Q\) and \(vQ\subseteq PP\) (Figure 6), one has \(vvx\subseteq PP\), so \(vvx\subseteq PP\), \(v\) is not a prefix of \(x\), and thus \(v=xv^{\prime}\) for some \(v^{\prime}\
eq\lambda\). Taking \(y\) and \(xy\), which are another pair of prefixes of \(Q\), we get \(x=ybx^{\prime}\) (\(x^{\prime}\) is possibly empty) because \(xxy\subseteq QQ\). Note that if \(v^{\prime}\) is a prefix of \(x\), then \((xv^{\prime})^{3}\subseteq xv^{\prime}xv^{\prime}xx\subseteq vQQ\subseteq PP\), which is impossible. Thus, \(v^{\prime}\) is not a prefix of \(x\) and then of \(y\). Since \(xv^{\prime}\) and \(xya\) are prefixes of \(Q\), we get \(v^{\prime}=yag\) for some (possibly empty) \(g\). Thus, \(Q=ybx^{\prime}yaybxb^{\prime}\), \(P=gybx^{\prime}ybx^{\prime}yagbxb^{\prime}ya\). We express \(p,q\), and \(l=q-|yb|\) in terms of \(|x^{\prime}|,|g|,|y|\):  
\[ \begin{array}{rcl}p&=&5(|y|+1)|+&3|x^{\prime}|+&2|g|,\\ q&=&3(|y|+1)|+&2|x^{\prime}|+&|g|,\\ l&=&2(|y|+1)|+&2|x^{\prime}|+&|g|.\end{array}  \tag{4}\]  
From (4), we get \(q=\frac{l+2p-|g|}{4}\leq\frac{l+2p}{4}\) and \(q=p-l+|x^{\prime}|\geq p-l\); together with the boundary \(q=l\), the obtained two lines form the green triangle in Zone IV with the vertices \((\frac{p}{2},\frac{p}{2})\), \((\frac{2}{3}p,\frac{3}{3}p)\), \((\frac{2}{3}p,\frac{2}{3}p)\) (Figure 2). Thus, we identified all \"red\" and \"green\" parts of the areas I-IV, getting the full picture from Figure 2. Theorem 2 is proved. The second crucial step in the proof of Theorem 2 is the following lemma on the density of positions fixed by cubes with similar periods. Figure 5: Location of factors of \(w\) for Zone III: \(q<p\), \(q\leq l\), \(3q>l+p\). The leftmost \(Q\) consists of a suffix of \(P\), followed by a shorter prefix of \(P\); the middle \(Q\) ends with some suffix \(y\) outside \(P^{\prime}\), possibly empty; \(P=zvxa\) is partitioned accordingly. Figure 6: Location of factors of \(w\) for Zone IV: \(q>p/2\), \(q>l\), \(2q<l+p\). The leftmost \(Q\) consists of a suffix of \(P\), followed by a shorter prefix of \(P\); the middle \(Q\) is a proper factor of \(P\); \(P=zvxa\) is partitioned accordingly. **Lemma 3**. : _Suppose that \(l\geq 1\), \(p\geq 2\) are integers, and \(w\) is a cube-free word such that \(|w|>l\). Among any \(l\) consecutive letters of \(w\), those less than \(\frac{8}{5}+\frac{3l}{5p}\) are fixed by cubes with periods in the range \([p..2p-1]\)._  
Proof. : Let us consider an inverse problem:  
* Let \(l_{0}<l_{1}<\cdots<l_{s}\) (\(s\geq 1\)) be positions in \(w\) containing letters fixed by cubes with periods \(q_{0},\ldots,q_{s}\), respectively, where \(q_{i}\in[p..2p-1]\) for all \(i\); find a lower bound for \(l_{s}-l_{0}\) (as a function of \(s\) and \(p\)) which applies for every sequence \(q_{0},\ldots,q_{s}\). The distance between each two consecutive position \(l_{i}\) and \(l_{i+1}\) is lower-bounded by Theorem 2. More precisely, we use Theorem 2 to make conclusions of the form
* \(l_{i+1}-l_{i}\geq A\), where the point \((A,q_{i+1})\) is on the border of the red polygon in Figure 2, in which \(p=q_{i}\), \(l=l_{i}\), and \(q=q_{i+1}\). For example, since the point \((25,15)\) is on the segment \(q=l-p\) of the border of such a polygon built for \(p=10\), we conclude that the condition \(q_{i}=10,q_{i+1}=15\) implies \(l_{i+1}-l_{i}\geq 25\). Let \(\beta=q_{i}\), \(\alpha=q_{i+1},l=l_{i+1}-l_{i}\).