On the regularity of the generalised golden ratio function

Abstract

International audienceGiven 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

Sturmian words and Cantor sets arising from unique expansions over ternary alphabets

“…That is to say, both (c) and (d) are consistent with the result of [12]. (ii) The above theorem also extends [2,Theorem 4]. Baker…”

“…At each kth step, C can be covered by n k intervals of length less than δ k := m 2 0 /2 k+2 . We havedim H C ≤ lim k→∞ log n k − log δ k ≤ lim k→∞ log(3k 2 /π 2 + O(k log k)) −2 log m 0 + (k + 2) log 2 = 0.After the author had obtained the whole work in this paper, he recognized in[2] the same result as Theorem 6.1. The current specific proof leads us to consider a sharper dimension function than that of the usual Hausdorff dimension.…”

“…For the sake of convenience and for coherence with (7), equations (2) and 3are restated in terms of the admissible sequence d.…”

“…So the results in it should be carefully translated. For instance, let β be such that d β (1) = 20(z 2,3 01) ∞ , or in other words, the unique real root >1 of the polynomial x 5 − 2x 4 − 2x 2 + 2x − 1 = 0. If m := 1 + 1/β(β − 2), then [16,Theorem 3.1] is no more true by Theorem 5.5 below.…”

“…But we have seen that this phenomenon cannot happen in the proof of Lemma 2.2.Proof of Theorem 4.6. From (β − 1)2 = m/(m − 1), we have m…”