On Small Bases for Which 1 Has Countably Many Expansions

Zou, Yuru; Wang, Lijin; Lü, Jian; Baker, Simon

doi:10.1112/s002557931500025x

Cited by 7 publications

(2 citation statements)

References 18 publications

(38 reference statements)

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“…, M}, where M is a given positive integer. See, e.g., [3,5,9,10,13,14,21,22,23] and their references. One of the main tools in these investigations is the lexicographic characterization of specific expansions; this fruitful idea was introduced by Parry [18].…”

Section: Introductionmentioning

confidence: 99%

Expansions in multiple bases over general alphabets

Zou¹,

Komornik²,

Lü³

2021

Preprint

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Expansions in non-integer bases have been extensively investigated since a pioneering work of Rényi. We introduce a more general framework of alphabet-base systems that also includes Pedicini's general alphabets and the multiple-base expansions of Neunhäuserer and Li. We extend the Parry type lexicographic theory to this setup, and we improve and generalize various former results on unique expansions.

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Section: Introductionmentioning

confidence: 99%

Expansions in multiple bases over general alphabets

Zou¹,

Komornik²,

Lü³

2021

Preprint

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“…This is well understood today for U 1 q [9,20], but the theory is far for complete if j ≥ 2. However, a number of important theorems have been obtained by Erdős et [12,33,6,4,5,27,36,37,19]. In order to mention some of these results concerning the two-digit case M = 1, let us denote by B j the set of bases q for which U j q is non-empty.…”

Section: Introductionmentioning

confidence: 99%

Univoque graphs and multiple expansions

Zou,

Lu,

Komornik

2019

Preprint

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Unique expansions in non-integer bases q have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description of the set of sequences generated by these graphs.Using the description of univoque graphs, the second part of the paper is devoted to the study of multiple expansions. Contrary to the unique expansions, we prove for each j ≥ 2 that the set U j q of numbers having exactly j expansions is closed only if it is empty.Furthermore, generalizing an important example of Sidorov [33], we prove for a large class of bases that the Hausdorff dimension of U j q is independent of j. In the last two sections our results are illustrated by many examples.

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