Generalized golden ratios of ternary alphabets

Komornik, Vilmos; Lai, Anna Chiara; Pedicini, Marco

doi:10.4171/jems/277

Cited by 23 publications

(31 citation statements)

References 13 publications

(28 reference statements)

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“…By the above G(m) = G( m m−1 ) and we may therefore assume m ∈ (1,2]. The authors of [4] considered m ≥ 2, and their results read as follows in our setting. In Section 3, we reprove all these results, making some of the statements more explicit and simplifying several proofs.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Remark 3.1. Komornik, Lai and Pedicini [4] observed that the sequences u ∈ S ∞ with the leading 0 removed are exactly the standard Sturmian sequences. However, they omitted the word "standard".…”

Section: 1 Statementsmentioning

confidence: 97%

“…Komornik, Lai and Pedicini [4] described the function m → G(m) on the interval (1,2]. We provide more details for this function, in particular for the set…”

Section: 1 Statementsmentioning

confidence: 99%

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On the regularity of the generalised golden ratio function

Baker

Steiner

2017

Bull. London Math. Soc.

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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

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: 1 Statementsmentioning

confidence: 97%

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On the regularity of the generalised golden ratio function

Baker

Steiner

2017

Bull. London Math. Soc.

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“…By an affine transformation it suffices to consider the alphabets {0, 1, m} with m ∈ [2, ∞). Properties (i)-(viii) below have been obtained in [17] (see Theorem 1.1, the proof of Lemma 5.3 and Remark 5.12); (ix) is due to Lai [23].…”

Section: Introductionmentioning

confidence: 91%

Critical bases for ternary alphabets

Komornik¹,

Pedicini²

2017

Acta Math. Hungar.

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“…Today's already classical definition of β-expansions was introduced in late 1950s [25,23] and this concept has up to now been a subject of active research, e.g. [1,3,4,5,7,8,9,10,12,18,19,20,24,26,27,28]. The world of β-expansions is much richer than that of the usual integer-base representations as is briefly illustrated on the uniqueness issue of β-expansions.…”

Section: β-Expansionsmentioning

confidence: 99%

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

Šíma¹

2017

mendel

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We briefly survey the basic concepts and results concerning the computational power of neural net-orks which basically depends on the information content of eight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classi ed within the Chomsky and finer complexity hierarchies. Then we re ne the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy.

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Generalized golden ratios of ternary alphabets

Cited by 23 publications

References 13 publications

On the regularity of the generalised golden ratio function

On the regularity of the generalised golden ratio function

Critical bases for ternary alphabets

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

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