  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 $\widetilde{\mathcal U}_\beta$.
