Critical base for the unique codings of fat Sierpinski gasket

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