On the regularity of the generalised golden ratio function

Abstract

Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcal{G}(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that $\mathcal{G}(A)$ varies continuously with the alphabet $A$ (of fixed size). What is more, we demonstrate that as we vary a single parameter $m$ within~$A$, the generalised golden ratio function may behave like $m^{1/h}$ for any positive integer $h$. These results follow from a detailed study of $\mathcal{G}(A)$ for ternary alphabets, building upon the work of Komornik, Lai, and Pedicini (2011). We give a new proof of their main result, that is we explicitly calculate the function $\mathcal{G}(\{0,1,m\})$. (For a ternary alphabet, it may be assumed without loss of generality that $A = \{0,1,m\}$ with $m\in(1,2)]$.) We also study the set of $m \in (1,2]$ for which $\mathcal{G}(\{0,1,m\})=1+\sqrt{m},$ we prove that this set is uncountable and has Hausdorff dimension~$0$. We show that the function mapping $m$ to $\mathcal{G}(\{0,1,m\})$ is of bounded variation yet has unbounded derivative. Finally, we show that it is possible to have unique expansions as well as points with precisely two expansions at the generalised golden ratio.