The authors deal with an invariant subset of the fat Sierpinski gasket \(S_{\beta}\) in \(\mathbb{R}^2\) generated by the iterated function system consisting of all points whose orbit never enters the overlap region \(\mathcal{O}_{\beta}\). The authors investigate the lexicographical characterization of the intrinsic univoque set \[ \widetilde{U}_{\beta}:=\left\{\sum_{i=1}^{\infty} \frac{d_i}{\beta_i}\in S_{\beta}:\sum_{i=1}^{\infty} \frac{d_{n+i}}{\beta_i}\notin \mathcal{O}_{\beta}\quad\forall n\geq 0\right\}. \] Denote the univoque set by \[ U_{\beta}:=\{P\in S_{\beta} : P \mathrm{\,has \, a \,unique \,coding \,with \,alphabet\,} \mathcal{A}\}.\]
Based on this characterization, the authors give an alternate proof of a theorem by N. Sidorov [Nonlinearity 20, No. 5, 1299–1312 (2007; Zbl 1122.60066)]. Furthermore, they prove a theorem which provides an affirmative answer to a conjecture in the above-mentioned paper.
Then the authors investigate all possible admissible words in \(\widetilde{U}_{\beta_c}\) based on the three types of Thue-Morse words with alphabet \(\mathcal{A}\).
Theorem. The number \(\beta_c\) is transcendental.
(1) If \(\beta \in (\beta_G, \beta_c)\), then \(\widetilde{U}_{\beta}\) is countably infinite;
(2) If \(\beta=\beta_c\), then \(\widetilde{U}_{\beta}\) is uncountable but has zero Hausdorff dimension;
(3) If \(\beta\in (\beta_c,2)\), then \(\widetilde{U}_{\beta}\) has positive Hausdorff dimension.
In the final section some questions are posed.