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

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.

Keywords

• full measure
• null measure
• lebesgue measure
• real line
• transcendental numbers
• liouville numbers