Critical base for the unique codings of fat Sierpinski gasket

Given β ∈ (1, 2) the fat Sierpinski gasket Sβ is the self-similar set in R2 generated by the iterated function system (IFS) fβ,d(x)=x+dβ,d∈A≔(0,0),(1,0),(0,1). Then for each point P∈Sβ there exists a sequence (di)∈AN such that P=∑i=1∞di/βi, and the infinite sequence (di) is called a coding of P. In general, a point in Sβ may have multiple codings since the overlap region Oβ≔⋃c,d∈A,c≠dfβ,c(Δβ)∩fβ,d(Δβ) has non-empty interior, where Δβ is the convex hull of Sβ. In this paper we are interested in the invariant set Ũβ≔∑i=1∞diβi∈Sβ:∑i=1∞dn+iβi∉Oβ∀n⩾0. Then each point in Ũβ has a unique coding. We show that there is a transcendental number βc ≈ 1.552 63 related to the Thue-Morse sequence, such that Ũβ has positive Hausdorff dimension if and only if β > βc. Furthermore, for β = βc the set Ũβ is uncountable but has zero Hausdorff dimension, and for β < βc the set Ũβ 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 Ũβ.