Displaying similar documents to “All Liouville Numbers are Transcendental”

Introduction to Liouville Numbers

Adam Grabowski, Artur Korniłowicz (2017)

Formalized Mathematics

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The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and [...] It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number....

A note on two linear forms

Nikolay Moshchevitin (2014)

Acta Arithmetica

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We prove a result on approximations to a real number θ by algebraic numbers of degree ≤ 2 in the case when we have certain information about the uniform Diophantine exponent ω̂ for the linear form x₀ + θx₁ + θ²x₂.

An extension of a theorem of Duffin and Schaeffer in Diophantine approximation

Faustin Adiceam (2014)

Acta Arithmetica

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Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended to the case where the numerators and the denominators of the rational approximants are related by a congruential constraint stronger than coprimality.