For a real base $\beta > 1$, there exist numbers with unique $\beta$-expansions on the alphabet $\{0,1\}$ if and only if $\beta$ is larger than the golden ratio. For ternary alphabets, the corresponding thresholds, called generalised golden ratios, were characterised by Komornik, Lai and Pedicini (2011), using Sturmian sequences. Another threshold is between bases with at most countably many unique expansions and bases with uncountably many unique expansions. For the alphabet $\{0,1\}$, Glendinning and Sidorov (2001) proved that this critical base is the Komornik-Loreti constant. For $\{0,1,2\}$, it was determined by de Vries and Komornik (2009). Since a complete characterisation for ternary alphabets seems to be difficult, we consider instead the question when uncountably many expansions using only digits in $\{0,1\}$ are unique with respect to the alphabet $\{0,1,m\}$, $m \in (1,2]$. Improving on results of Komornik and Pedicini (2017), we give a complete characterisation of the corresponding critical bases, using $S$-adic sequences that form a mixture of Sturmian sequences and the Thue-Morse sequence.
