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Zero-full law for well approximable sets in missing digit sets

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  20 February 2025

BING LI ,
RUOFAN LI  and
YUFENG WU
BING LI
Affiliation:
School of Mathematics, South China University of Technology, Guangzhou, China, 510641. e-mail: [email protected]
RUOFAN LI
Affiliation:
Department of Mathematics, Jinan University, Guangzhou, China, 510632. e-mail: [email protected]
YUFENG WU
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, China, 410083. e-mail: [email protected]
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Abstract

Let $b \geqslant 3$ be an integer and C(b, D) be the set of real numbers in [0,1] whose base b expansion only consists of digits in a set $D {\subseteq} \{0,...,b-1\}$. We study how close can numbers in C(b, D) be approximated by rational numbers with denominators being powers of some integer t and obtain a zero-full law for its Hausdorff measure in several circumstances. When b and t are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann. 338 (2007), 97–118) and generalise their theorem. When b and t are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11J82: Measures of irrationality and of transcendence 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 178 , Issue 1 , January 2025 , pp. 81 - 102
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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