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Metric results for numbers with multiple q-expansions

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posted on 2023-12-14, 16:40 authored by Simon BakerSimon Baker, Yuru Zou

Let M be a positive integer and q ∈ (1,M +1]. A q-expansion of a real number x is a sequence (ci) = c1c2 · · · with ci ∈ {0, 1, . . . ,M} such that x = [equation here]. In this paper we study the set Ujq consisting of those real numbers having exactly j q-expansions. Our main result is that for Lebesgue almost every q ∈ (qKL,M + 1), we have

dimH Ujq ≤ max{0, 2 dimH Uq − 1} for all j ∈ {2, 3, . . .}.

Here qKL is the Komornik-Loreti constant. As a corollary of this result, we show that for any j ∈ {2, 3, . . .}, the function mapping q to dimH Ujq is not continuous.

Funding

Fractal Properties of Some Sets in Real Non-Integer Base Representation and Related Problems

National Natural Science Foundation of China

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Research on High-dimensional Image Restoration Model and Algorithm Based on Low Rank Tensor

National Natural Science Foundation of China

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Shenzhen Basis Research Project (JCYJ20210324094006017)

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Fractal Geometry

Volume

10

Issue

3/4

Pages

243–266

Publisher

European Mathematical Society (EMS)

Version

  • VoR (Version of Record)

Rights holder

© European Mathematical Society

Publisher statement

This is an Open Access article published by EMS Press. This work is licensed under a CC BY 4.0 license, https://creativecommons.org/licenses/by/4.0/.

Acceptance date

2023-02-19

Publication date

2023-08-10

Copyright date

2023

ISSN

2308-1309

eISSN

2308-1317

Language

  • en

Depositor

Dr Simon Baker. Deposit date: 24 February 2023

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