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### Liouville numbers, Liouville sets and Liouville fields

K. Senthil Kumar, R. Thangadurai, M. Waldschmidt
2015 Proceedings of the American Mathematical Society
Maillet 100 years ago, we introduce the definition of a Liouville set, which extends the definition of a Liouville number.  ...  Any Liouville number belongs to a Liouville set S having the power of continuum and such that Q ∪ S is a Liouville field. Update: May 25, 2013  ...  number or a Liouville number, and in the second case S ∪ {η} is a Liouville set.  ...

### Liouville numbers, Liouville sets and Liouville fields [article]

K. Senthil Kumar and R. Thangadurai and M. Waldschmidt
2013 arXiv   pre-print
Any Liouville number belongs to a Liouville set S having the power of continuum and such that the union of S with the rational number field is a Liouville field.  ...  Following earlier work by E.Maillet 100 years ago, we introduce the definition of a Liouville set, which extends the definition of a Liouville number.  ...  rational number or a Liouville number, and in the second case S ∪ {η} is a Liouville set.  ...

### On strong liouville numbers

G. Petruska
1992 Indagationes mathematicae
It is shown that sums and products of strong Liouville numbers are Liouville numbers (or rationals), but usually not strong Liouville numbers.  ...  Also, the classical Liouville numbers defined by infinite series are not strong Liouville numbers.  ...  The sum or the product of an arbitrary number of strong Liouville numbers is either a rational or a Liouville number.  ...

### Introduction to Liouville Numbers

2017 Formalized Mathematics
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.  ...  The aim is to show that all Liouville numbers are transcendental.  ...  One can check that LiouvilleConst is Liouville and there exists a real number which is Liouville. )(i) ((c − 1) · pfact(b))(i). (38) A Liouville number is a Liouville real number.  ...

### On approximation constants for Liouville numbers

Johannes Schleischitz
2015 Glasnik Matematicki - Serija III
. , ζ k ) for Liouville numbers ζ. For a certain class of Liouville numbers including the famous representative n≥1 10 −n!  ...  and numbers in the Cantor set, we explicitly determine all approximation constants simultaneously for all k ≥ 1.  ...  The concern of the first theorem is to determine/bound the classic approximation constants for arbitrary Liouville numbers. Theorem 3.1. Let ζ be a Liouville number.  ...

### On Approximation constants for Liouville numbers [article]

Johannes Schleischitz
2015 arXiv   pre-print

### Periodic distributions and non-Liouville numbers

Gary H Meisters
1977 Journal of Functional Analysis
be Liouville numbers and there are only a denumerable number of algebraic numbers.  ...  Also it follows t from the definition of Liouville numbers (but more immediately from Theorem 1) that the reciprocal of a non-Liouville number is also a non- Liouville number.  ...

### A computable absolutely normal Liouville number

Verónica Becher, Pablo Ariel Heiber, Theodore A. Slaman
2015 Mathematics of Computation
There is a computable absolutely normal Liouville number.  ...  , is the standard example of a Liouville number. Though uncountable, the set of Liouville numbers is small, in fact, it is null, both in Lebesgue measure and in Hausdorff dimension (see [2] ).  ...