Critical base for the unique codings of fat Sierpinski gasket

Given $\beta\in(1,2)$ the fat Sierpinski gasket $\mathcal S_\beta$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) $$f_{\beta,d}(x)=\frac{x+d}{\beta},\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}.$$ Then for each point $P\in\mathcal S_\beta$ there exists a sequence $(d_i)\in\mathcal A^\mathbb N$ such that $P=\sum_{i=1}^\infty d_i/\beta^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in $\mathcal S_\beta$ may have multiple codings since the overlap region $\mathcal O_\beta:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{\beta,c}(\Delta_\beta)\cap f_{\beta,d}(\Delta_\beta)$ has non-empty interior, where $\Delta_\beta$ is the convex hull of $\mathcal S_\beta$. In this paper we are interested in the invariant set $$\widetilde{\mathcal U}_\beta:=\left\{\sum_{i=1}^\infty \frac{d_i}{\beta^i}\in \mathcal S_\beta: \sum_{i=1}^\infty\frac{d_{n+i}}{\beta^i}\notin\mathcal O_\beta~\forall n\ge 0\right\}.$$ Then each point in $\widetilde{\mathcal U}_\beta$ has a unique coding. We show that there is a transcendental number $\beta_c\approx 1.55263$ related to the Thue-Morse sequence, such that $\widetilde{\mathcal U}_\beta$ has positive Hausdorff dimension if and only if $\beta>\beta_{c}$. Furthermore, for $\beta=\beta_c$ the set $\widetilde{\mathcal U}_\beta$ is uncountable but has zero Hausdorff dimension, and for $\beta<\beta_c$ the set $\widetilde{\mathcal U}_\beta$ is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of $\widetilde{\mathcal U}_\beta$.