Topological and dimensional properties of univoque bases in double-base expansions

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-24 DOI:10.1016/j.topol.2025.109294

Yuecai Hu , Rafael Alcaraz Barrera , Yuru Zou

引用次数: 0

Abstract

Given two real numbers

q_{0}, q_{1}

with

q_{0}, q_{1} > 1

satisfying

q_{0} + q_{1} \geq q_{0} q_{1}

, we call a sequence

(d_{i})

with

d_{i} \in {0, 1}

(q_{0}, q_{1})

-expansion or a double-base expansion of a real number x if

x = \sum_{i = 1}^{\infty} \frac{d_{i}}{q_{d_{1}} q_{d_{2}} \dots q_{d_{i}}} .

When

q_{0} = q_{1} = q

, the set of univoque bases is given by the set of q's such that

x = 1

has exactly one

(q, q)

-expansion. The topological, dimensional and symbolic properties of such sets and their corresponding sequences have been intensively investigated. In our research, we study the topological and dimensional properties of the set of univoque bases for double-base expansions. This problem is more complicated, requiring new research strategies. Several new properties are uncovered. In particular, we show that the set of univoque bases in the double base setting is a meagre set with full Hausdorff dimension.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.