# 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.
Type de document :
Article dans une revue
Bulletin of the London Mathematical Society, London Mathematical Society, 2017, 49 (1), pp.58 - 70. 〈10.1112/blms.12002〉
Domaine :

https://hal.archives-ouvertes.fr/hal-01620763
Contributeur : Wolfgang Steiner <>
Soumis le : vendredi 20 octobre 2017 - 23:11:07
Dernière modification le : jeudi 11 janvier 2018 - 06:27:38

### Citation

Simon Baker, Wolfgang Steiner. On the regularity of the generalised golden ratio function. Bulletin of the London Mathematical Society, London Mathematical Society, 2017, 49 (1), pp.58 - 70. 〈10.1112/blms.12002〉. 〈hal-01620763〉

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