Liouville numbers and Schanuel’s Conjecture

Kumar, K. Senthil; Thangadurai, R.; Waldschmidt, Michel

doi:10.1007/s00013-013-0606-0

Cited by 14 publications

(23 citation statements)

References 7 publications

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“…Recall a Liouville number is an irrational real (and thus transcendental by Liouville's Theorem) number that satisfies λ 1 (ζ) = ∞. It is shown in [9] that for any countable set of continuous strictly monotonic functions f i : A → B with A, B non-empty intervals of R, there are uncountably many Liouville numbers ζ ∈ A such that f i (ζ) is again a Liouville number for all i. See also [13], [16].…”

Section: A New Exponent Of Simultaneous Approximationmentioning

confidence: 99%

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

Schleischitz

2016

Monatsh Math

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This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and encourages to define a new exponent of simultaneous approximation. The second part of the paper is devoted to the study of this exponent.

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Section: A New Exponent Of Simultaneous Approximationmentioning

confidence: 99%

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

Schleischitz

2016

Monatsh Math

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“…If we weaken this to ζ ∈ L , we further have no uniform bound in (19) which is needed to bound the left hand side in (21), even restricting to ζ in a given compact interval. A negative answer can be readily inferred from a recent result on Liouville numbers [13], which bases solely on the topological property of L being a G δ dense set. See also [2], [10], [24], [29] and [5] (however, as pointed out in the MathSciNet review, the proof in [5] has a small gap and it does not work in general.…”

Section: 2mentioning

confidence: 99%

On a Problem Posed by Mahler

Marques¹,

Schleischitz²

2015

J. Aust. Math. Soc.

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Abstract. E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to f (z) = z q where q is a rational number.

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“…Indeed, if ξ is a strong Liouville number, then F(ξ ) is a U mnumber, for all rational function F(x) with coefficients in an m-degree number field. For more about this subject, we refer the reader to the recent works (Senthil Kumar 2015; Senthil Kumar et al 2014) and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the Arithmetic Behavior of Liouville Numbers Under Rational Maps

Chaves

Marques

Trojovský

2020

Bull Braz Math Soc, New Series

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In 1972, Alniaçik proved that every strong Liouville number is mapped into the set of U m-numbers, for any non-constant rational function with coefficients belonging to an m-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense).

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Liouville numbers and Schanuel’s Conjecture

Abstract: In this paper, using an argument of P. Erdős, K. Alniaçik and E. Saias, we extend earlier results on Liouville numbers, due to P. Erdős,

Cited by 14 publications

References 7 publications

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

On a Problem Posed by Mahler

On the Arithmetic Behavior of Liouville Numbers Under Rational Maps

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