Number theoretic subsets of the real line of full or null measure

Taboka Prince Chalebgwa; Sidney A. Morris; Taboka Prince Chalebgwa; Sidney A. Morris

doi:10.3934/era.2025046

Electronic Research Archive

2025, Volume 33, Issue 2: 1037-1044. doi: 10.3934/era.2025046

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Expository

Number theoretic subsets of the real line of full or null measure

Taboka Prince Chalebgwa ^1,2,
Sidney A. Morris ^{3,4
,
,}

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa
2.
National Institute for Theoretical and Computational Sciences (NiTheCS), South Africa
3.
Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Victoria 3086, Australia
4.
School of Engineering, IT and Physical Sciences, Federation University Australia, PO Box 663, Ballarat, Victoria 3353, Australia

Received: 03 December 2024 Revised: 19 February 2025 Accepted: 19 February 2025 Published: 25 February 2025

During a first or second course in number theory, students soon encounter several sets of "number theoretic interest". These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent $ m $, among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.
- full measure,
- null measure,
- Lebesgue measure,
- real line,
- transcendental numbers,
- Liouville numbers
Citation: Taboka Prince Chalebgwa, Sidney A. Morris. Number theoretic subsets of the real line of full or null measure[J]. Electronic Research Archive, 2025, 33(2): 1037-1044. doi: 10.3934/era.2025046

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Abstract

During a first or second course in number theory, students soon encounter several sets of "number theoretic interest". These include basic sets such as the rational numbers, algebraic numbers, transcendental numbers, and Liouville numbers, as well as more exotic sets such as the constructible numbers, normal numbers, computable numbers, badly approximable numbers, the Mahler sets S, T and U, and sets of irrationality exponent $ m $, among others. Those exposed to some measure theory soon make a curious observation regarding a common property seemingly shared by all these sets: each of the sets has Lebesgue measure equal to zero, or its complement has Lebesgue measure equal to zero. In this expository note, we explain this phenomenon.

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