arXiv subjects

Anthony Poëls

Publications and source records attributed to Anthony Poëls.

13 recordsLinked to original sources

arXiv2024

On the linear independence of $p$-adic polygamma values

In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $ζ_p(s,x)$ at distinct shifts $x$. This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. Our proof is based on a novel and explicit construction of Padé-type approximants of the second kind of Diamond's $p$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

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arXiv2024

On approximation to a real number by algebraic numbers of bounded degree

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $ξ$ can be approximated by algebraic numbers $α$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(α)$ of $α$. He showed that the infimum $ω^*_n(ξ)$ of all $ω$ for which infinitely many such $α$ have $|ξ-α| \le H(α)^{-ω-1}$ is at least $(n+1)/2$. He also asked if we could even have $ω^*_n(ξ) \ge n$ as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form $n/2+\mathcal{O}(1)$ until Badziahin and Schleischitz showed in 2021 that $ω^*_n(ξ) \ge an$ for each $n\ge 4$, with $a=1/\sqrt{3}\simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $ω^*_n(ξ) \ge an$ for each $n\ge 2$, with $a=1/(2-\log 2)\simeq 0.765$.

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arXiv2024

On uniform polynomial approximation

Let $n$ be a positive integer and $ξ$ a transcendental real number. We are interested in bounding from above the uniform exponent of polynomial approximation $\widehatω_n(ξ)$. Davenport and Schmidt's original 1969 inequality $\widehatω_n(ξ)\leq 2n-1$ was improved recently, and the best upper bound known to date is $2n-2$ for each $n\geq 10$. In this paper, we develop new techniques leading us to the improved upper bound $2n-\frac{1}{3}n^{1/3}+\mathcal{O}(1)$.

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arXiv2023

Parametric geometry of numbers over a number field and extension of scalars

The parametric geometry of numbers of Schmidt and Summerer deals with rational approximation to points in $\mathbb{R}^n$. We extend this theory to a number field $K$ and its completion $K_w$ at a place $w$ in order to treat approximation over $K$ to points in $K_w^n$. As a consequence, we find that exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative degree one over a place $\ell$ of $\mathbb{Q}$, we further relate approximation over $K$ to a point $\boldsymbolξ$ in $K_w^n$, to approximation over $\mathbb{Q}$ to a point $Ξ$ in $\mathbb{Q}_\ell^{nd}$, obtained by extension of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination with a result of Bel, this allows us to construct algebraic curves in $\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points that are very singular with respect to rational approximation.

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arXiv2022

$S$-unit equation in two variables and Padé approximations

In this article, we use Padé approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Padé approximants with a simple argument relying on Mahler measure and on the local height, we refine the bound due to J.-H. Evertse.

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arXiv2022

Simultaneous rational approximation to successive powers of a real number

We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehatλ_n(ξ)$ to successive powers $1,ξ,\dots,ξ^n$ of a given real transcendental number $ξ$. As an application, we obtain a refined lower bound for the exponent of approximation to $ξ$ by algebraic integers of degree at most $n+1$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.

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arXiv2022

Padé approximation for a class of hypergeometric functions and parametric geometry of numbers

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Padé approximations by using a formal method and show that the associated sequences satisfy a Poincaré-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an \emph{effective} version of the Poincaré-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers.

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arXiv2021

Exponents of Diophantine approximation in dimension two for a general class of numbers

We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We compute, for each real number $ξ$ of this set, several exponents of Diophantine approximation to the pair $(ξ,ξ^2)$, together with $ω_2^*(ξ)$ and $\widehatω_2^*(ξ)$, the so-called ordinary and uniform exponent of approximation to $ξ$ by algebraic numbers of degree $\leq 2$. As an application, we get new information on the set of values taken by $\widehatω_2^*$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent $β_0$.

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arXiv2020

A class of maximally singular sets for rational approximation

We say that a subset of $\mathbb{P}^n(\mathbb{R})$ is maximally singular if its contains points with $\mathbb{Q}$-linearly independent homogenous coordinates whose uniform exponent of simultaneous rational approximation is equal to $1$, 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.

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arXiv2019

Rational approximation to real points on quadratic hypersurfaces

Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/ρ$ of an explicit Pisot number $ρ<2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound $1/ρ$ uses a recent transference inequality of Marnat and Moshchevitin. In the case $n=3$, we recover results of the second author while for $n>3$, this completes recent work of Kleinbock and Moshchevitin.

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arXiv2019

A transference principle for simultaneous rational approximation

We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.

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arXiv2019

A new exponent of simultaneous rational approximation

We introduce a new exponent of simultaneous rational approximation $\widehatλ_{\min}(ξ,η)$ for pairs of real numbers $ξ,η$, in complement to the classical exponents $λ(ξ,η)$ of best approximation, and $\widehatλ(ξ,η)$ of uniform approximation. It generalizes Fischler's exponent $β_0(ξ)$ in the sense that $\widehatλ_{\min}(ξ,ξ^2) = 1/β_0(ξ)$ whenever $λ(ξ,ξ^2) = 1$. Using parametric geometry of numbers, we provide a complete description of the set of values taken by $(λ,\widehatλ_{\min})$ at pairs $(ξ,η)$ with $1$, $ξ$, $η$ linearly independent over $\mathbf{Q}$.

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arXiv2017

Exponents of diophantine approximation in dimension $2$ for numbers of Sturmian type

We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $ω_2(ξ)$, $\widehatω_2(ξ)$, $λ_2(ξ)$, $\widehatλ_2(ξ)$ associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.

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