scholarly journals A transference principle for simultaneous rational approximation

2020 ◽  
Vol 32 (2) ◽  
pp. 387-402
Author(s):  
Ngoc Ai Van Nguyen ◽  
Anthony Poëls ◽  
Damien Roy
Keyword(s):  
Rational Approximation ◽  
Transference Principle ◽  
Simultaneous Rational Approximation

scholarly journals On Infinitely Many Rational Approximants to ζ(3)

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1176 ◽  
Author(s):  
Jorge Arvesú ◽  
Anier Soria-Lorente
Keyword(s):  
Difference Equation ◽  
Rational Approximation ◽  
Difference Equations ◽  
Second Order ◽  
Rational Approximants ◽  
Linearly Independent ◽  
Approximation Problems ◽  
Simultaneous Rational Approximation

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers, II

2021 ◽  
pp. 1-21
Author(s):  
Dzmitry Badziahin ◽  
Yann Bugeaud ◽  
Johannes Schleischitz
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Simultaneous Rational Approximation

Abstract Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$ , let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$ , where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$ . We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

scholarly journals A class of maximally singular sets for rational approximation

2020 ◽  
Vol 16 (09) ◽  
pp. 2005-2012
Author(s):  
Anthony Poëls
Keyword(s):  
Rational Approximation ◽  
Linearly Independent ◽  
Singular Sets ◽  
Quadratic Hypersurfaces ◽  
Simultaneous Rational Approximation

We say that a subset of [Formula: see text] is maximally singular if its contains points with [Formula: see text]-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to [Formula: see text], the maximal possible value. In this paper, we give a criterion which provides many such sets including Grassmannians. We also recover a result of the author and Roy about a class of quadratic hypersurfaces.

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers

2011 ◽  
Vol 54 (3) ◽  
pp. 599-612 ◽  
Author(s):  
Yann Bugeaud ◽  
Natalia Budarina ◽  
Detta Dickinson ◽  
Hugh O'Donnell
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Positive Integers ◽  
Simultaneous Rational Approximation

AbstractLet p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

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