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Liouville numbers, Liouville sets and Liouville fields

2015
*
Proceedings of the American Mathematical Society
*

Maillet 100 years ago, we introduce the definition of a

doi:10.1090/proc/12408
fatcat:pt374fhshjht5ii2sqcr5sr72y
*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. ...##
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Liouville numbers, Liouville sets and Liouville fields
[article]

2013
*
arXiv
*
pre-print

Any

arXiv:1312.7151v1
fatcat:dmvcizzfh5db5gmoqn7ikpgfdu
*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. ...##
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On strong liouville numbers

1992
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Indagationes mathematicae
*

It is shown that sums and products of strong

doi:10.1016/0019-3577(92)90010-i
fatcat:xpk7feoiwnbshlevp3nbu3voxq
*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*. ...##
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Introduction to Liouville Numbers

2017
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Formalized Mathematics
*

A real

doi:10.1515/forma-2017-0003
fatcat:w4nyxx4nubgghbjict34tcyfxu
*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*. ...##
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On approximation constants for Liouville numbers

2015
*
Glasnik Matematicki - Serija III
*

. , ζ k ) for

doi:10.3336/gm.50.2.06
fatcat:aqackebg6nae3ezpi7hi6l4a7y
*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*. ...##
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On Approximation constants for Liouville numbers
[article]

2015
*
arXiv
*
pre-print

For a certain class of

arXiv:1409.1396v3
fatcat:wtwvf5qbdvb2njkoysjn2gy3p4
*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*. ...##
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Global Hypoellipticity and Liouville Numbers

1972
*
Proceedings of the American Mathematical Society
*

It is now clear that (LM) is equivalent to a not a

doi:10.2307/2038523
fatcat:ubbu2li2srazzp57fndu4xtfyu
*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. ...##
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Global hypoellipticity and Liouville numbers

1972
*
Proceedings of the American Mathematical Society
*

It is now clear that (LM) is equivalent to a not a

doi:10.1090/s0002-9939-1972-0296508-5
fatcat:mptbssyo5rgqnpohkvcyw6lazy
*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. ...##
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Liouville numbers and Schanuel's Conjecture

2014
*
Archiv der Mathematik
*

Saias, we extend earlier results on

doi:10.1007/s00013-013-0606-0
fatcat:62vfuoy6ijeytmhwlybsooj4yq
*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*. ...##
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All Liouville Numbers are Transcendental

2017
*
Formalized Mathematics
*

A real

doi:10.1515/forma-2017-0004
fatcat:6fizkkpwmjgola45e5wtkljqgq
*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*. ...##
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Liouville Numbers and Schanuel's Conjecture
[article]

2013
*
arXiv
*
pre-print

Saias, we extend earlier results on

arXiv:1312.7154v1
fatcat:qj4djj66jfewhe2cqfkwong44a
*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*. ...##
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Diophantine approximation and special Liouville numbers
[article]

2013
*
arXiv
*
pre-print

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable

arXiv:1301.2177v1
fatcat:oesvxm7n4bgyjmrs4huy7qs36m
*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 ...##
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Periodic distributions and non-Liouville numbers

1977
*
Journal of Functional Analysis
*

be

doi:10.1016/0022-1236(77)90016-7
fatcat:gc7dvccbozbhfecgh4hun657ga
*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*. ...##
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A computable absolutely normal Liouville number

2015
*
Mathematics of Computation
*

There is a computable absolutely normal

doi:10.1090/mcom/2964
fatcat:b54wnhteercsbddxmnhm2ac5ci
*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] ). ...##
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Normality and Finite-state Dimension of Liouville numbers
[article]

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*. ...

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