-   the beginnings of long blocks are at a distance at least $8
    \log_2n$, in the sense $\mathcal G_n \subset D_n^c$;

-   the word $w$ has no equal long blocks, in the sense $\mathcal G_n \subset E_n^c$.

Note that on $\mathcal G_n$, the length of a long block is less than $3\log_2n+2\times2\log_2n= 7 \log_2{n}$, and that there is no overlapping between two long blocks . The next lemma ensures that a large proportion of Lyndon words are good Lyndon words:

\[Gn\] For every $n$ large enough: $${\tilde{\mathbb P}_n}(\mathcal G_n) \geq
1- O \left( n^{-(1-2\varepsilon)}\log_2 n \right).$$

**Proof.** Everything has been proved in the previous lemmas \[MnS\], \[lemMnGV\] (in which $A=2$), \[lemHn\], \[En\] and \[Dn\].

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Now, we note that a good Lyndon word $w\in
\mathcal G_n $ begins with a long block, ends with a run of “$b$”, and all portions between long blocks begin with a run of “$a$” and end with a run of “$b$”. We can thus give the following definition of short blocks:

For a good Lyndon word $w\in
\mathcal G_n $, we cut each section stretching between two long blocks into short blocks, made of two consecutive runs of “$a$” and “$b$”(in this order).

Note 