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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.  ... 
doi:10.1090/proc/12408 fatcat:pt374fhshjht5ii2sqcr5sr72y

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.  ... 
arXiv:1312.7151v1 fatcat:dmvcizzfh5db5gmoqn7ikpgfdu

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.  ... 
doi:10.1016/0019-3577(92)90010-i fatcat:xpk7feoiwnbshlevp3nbu3voxq

Introduction to Liouville Numbers

Adam Grabowski, Artur Korniłowicz
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.  ... 
doi:10.1515/forma-2017-0003 fatcat:w4nyxx4nubgghbjict34tcyfxu

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.  ... 
doi:10.3336/gm.50.2.06 fatcat:aqackebg6nae3ezpi7hi6l4a7y

On Approximation constants for Liouville numbers [article]

Johannes Schleischitz
2015 arXiv   pre-print
For a certain class of Liouville numbers including the famous representative $\sum_{n\geq 1} 10^{-n!}  ...  We investigate some Diophantine approximation constants related to the simultaneous approximation of $(\zeta,\zeta^{2},\ldots,\zeta^{k})$ for Liouville numbers $\zeta$.  ...  Roth's Theorem [10] asserts µ(ζ) = 2 for all algebraic irrational real numbers ζ. Irrational real numbers ζ with µ(ζ) = ∞ are called Liouville numbers.  ... 
arXiv:1409.1396v3 fatcat:wtwvf5qbdvb2njkoysjn2gy3p4

Global Hypoellipticity and Liouville Numbers

Stephen J. Greenfield, Nolan R. Wallach
1972 Proceedings of the American Mathematical Society  
It is now clear that (LM) is equivalent to a not a Liouville number. Remark.  ...  We recall (see [HW] ) that a e R is a Liouville number if it can be approximated by rationals to any order.  ... 
doi:10.2307/2038523 fatcat:ubbu2li2srazzp57fndu4xtfyu

Global hypoellipticity and Liouville numbers

Stephen J. Greenfield, Nolan R. Wallach
1972 Proceedings of the American Mathematical Society  
It is now clear that (LM) is equivalent to a not a Liouville number. Remark.  ...  We recall (see [HW] ) that a e R is a Liouville number if it can be approximated by rationals to any order.  ... 
doi:10.1090/s0002-9939-1972-0296508-5 fatcat:mptbssyo5rgqnpohkvcyw6lazy

Liouville numbers and Schanuel's Conjecture

K. Senthil Kumar, R. Thangadurai, M. Waldschmidt
2014 Archiv der Mathematik  
Saias, we extend earlier results on Liouville numbers, due to P. Erdős,  ...  Then all numbers of the sequence (ξ n ) n≥0 are Liouville numbers. (iv) For any rational number r = 0, the number ξ r is a Liouville number.  ...  Then there exists an uncountable set of Liouville numbers ξ ∈ I such that ϕ(ξ) is a Liouville number.  ... 
doi:10.1007/s00013-013-0606-0 fatcat:62vfuoy6ijeytmhwlybsooj4yq

All Liouville Numbers are Transcendental

Artur Korniłowicz, Adam Naumowicz, Adam Grabowski
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.  ...  Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated "quite closely" by a sequence of rational numbers.  ...  Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated "quite closely" by a sequence of rational numbers.  ... 
doi:10.1515/forma-2017-0004 fatcat:6fizkkpwmjgola45e5wtkljqgq

Liouville Numbers and Schanuel's Conjecture [article]

K. Senthil Kumar and R. Thangadurai and M. Waldschmidt
2013 arXiv   pre-print
Saias, we extend earlier results on Liouville numbers, due to P. Erdos, G.J. Rieger, W. Schwarz, K. Alniacik, E. Saias, E.B. Burger.  ...  We also produce new results of algebraic independence related with Liouville numbers and Schanuel's Conjecture, in the framework of G delta-subsets.  ...  Then all numbers of the sequence (ξ n ) n≥0 are Liouville numbers. (iv) For any rational number r = 0, the number ξ r is a Liouville number.  ... 
arXiv:1312.7154v1 fatcat:qj4djj66jfewhe2cqfkwong44a

Diophantine approximation and special Liouville numbers [article]

Johannes Schleischitz
2013 arXiv   pre-print
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$.  ...  As an application, explicit construction of real numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.  ...  Liouville numbers, that is real numbers ζ for which the inequality ζ − p q ≤ 1 q η has infinitely many rational solutions p q for arbitrarily large η ∈ R, will be suitable examples since they all satisfy  ... 
arXiv:1301.2177v1 fatcat:oesvxm7n4bgyjmrs4huy7qs36m

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.  ... 
doi:10.1016/0022-1236(77)90016-7 fatcat:gc7dvccbozbhfecgh4hun657ga

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] ).  ... 
doi:10.1090/mcom/2964 fatcat:b54wnhteercsbddxmnhm2ac5ci

Normality and Finite-state Dimension of Liouville numbers [article]

Satyadev Nandakumar, Santhosh Kumar Vangapelli
2014 arXiv   pre-print
Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers.  ...  This refines Staiger's result that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state  ...  A Normal Liouville Number Though the Liouville numbers constructed above were non-normal, there are normal Liouville numbers.  ... 
arXiv:1204.4104v2 fatcat:ip5u7lyuevdojoegq2xxdtpxia
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