Critical base for the unique codings of fat Sierpinski gasket

Given $β\in(1,2)$ the fat Sierpinski gasket $\mathcal S_β$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) \[ f_{β,d}(x)=\frac{x+d}β,\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}. \] Then for each point $P\in\mathcal S_β$ there exists a sequence $(d_i)\in\mathcal A^\mathbb N$ such that $P=\sum_{i=1}^\infty d_i/β^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in $\mathcal S_β$ may have multiple codings since the overlap region $\mathcal O_β:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{β,c}(Δ_β)\cap f_{β,d}(Δ_β)$ has non-empty interior, where $Δ_β$ is the convex hull of $\mathcal S_β$. In this paper we are interested in the invariant set \[ \widetilde{\mathcal U}_β:=\left\{\sum_{i=1}^\infty \frac{d_i}{β^i}\in \mathcal S_β: \sum_{i=1}^\infty\frac{d_{n+i}}{β^i}\notin\mathcal O_β~\forall n\ge 0\right\}. \] Then each point in $ \widetilde{\mathcal U}_β$ has a unique coding. We show that there is a transcendental number $β_c\approx 1.55263$ related to the Thue-Morse sequence, such that $\widetilde{\mathcal U}_β$ has positive Hausdorff dimension if and only if $β>β_{c}$. Furthermore, for $β=β_c$ the set $\widetilde{\mathcal U}_β$ is uncountable but has zero Hausdorff dimension, and for $β<β_c$ the set $\widetilde{\mathcal U}_β$ 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 $\widetilde{\mathcal U}_β$.