a_\ell] \cap \pi [\overline{a_i}] = {\varnothing}$;

-   $\pi [c_j c_{j+1} \dots c_n a_1 a_2 \dots a_\ell] \cap \pi [\overline{c_j}] = {\varnothing}$.

Then the images of $\{uv, uw\}^*$ under $\pi$ all have unique addresses. That is, the images of all infinite words of the form $t_1 t_2 t_3 \dots$ with $t_i \in \{uv, uw\}$ under $\pi$ all have unique addresses.

We see that any shift of a word from $\{uv, uw\}^*$ is such that it’s prefix will be of one of the four forms listed above. Further, by assumption, the first term is uniquely determined. By applying $T_m^{-1}$ or $T_p^{-1}$ as appropriate, we get that all terms are uniquely determined, which proves the result.

If the conditions of Lemma \[thm:unique tool\] are satisfied and $\{uv, uw\}^*$ is unambiguous, then $\dim_H\mathcal U>0$.

We recall that $\{uv, uw\}$ is ambiguous if there exists two sequences $(t_1, t_2, t_3, \dots) \neq (s_1, s_2, s_3, \dots)$ with $t_i, s_i \in \{uv, uw\}$ where $t_1 t_2 t_3 \dots = s_1 s_2 s_3 \dots$. If no such sequence exists, then this language is unambiguous. For example, $\{mpmp, mp\}^*$ would be ambiguous, whereas $\{m, pp\}^*$ would be unambiguous.

This is completely analogous to [@HS Corollary 4.3]. We say 