Exponents of Diophantine approximation in dimension 2 for a general class of numbers

Poëls, Anthony

doi:10.2140/moscow.2022.11.37

Cited by 2 publications

(2 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is known that for

n=2

\widehat{\lambda }_2(\xi )

can change between

\frac{1}{2}

and

\frac{\sqrt {5}-1}{2}

where both upper and lower bounds are sharp [11]. Recent developments about the spectrum of values

\widehat{\lambda }_2(\xi )

and

\widehat{\omega }_2(\xi )

can be found in [4, 9]. However, for

n\geqslant 3

and transcendental ξ, it is not even known if

\widehat{\lambda }_n(\xi )

can take any other value than

1/n

.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that for 𝑛 = 2, λ2 (𝜉) can change between 1 2 and √ 5−1 2 where both upper and lower bounds are sharp [11]. Recent developments about the spectrum of values λ2 (𝜉) and ω2 (𝜉) can be found in [4,9]. However, for 𝑛 ⩾ 3 and transcendental 𝜉, it is not even known if λ𝑛 (𝜉) can take any other value than 1∕𝑛.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Upper bounds for the uniform simultaneous Diophantine exponents

Badziahin

2022

Mathematika

View full text Add to dashboard Cite

We give several upper bounds for the uniform simultaneous Diophantine exponent λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ of a transcendental number ξ∈double-struckR$\xi \in \mathbb {R}$. The most important one relates λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ and the ordinary simultaneous exponent ωk(ξ)$\omega _k(\xi )$ in the case when k is substantially smaller than n. In particular, in the generic case ωk(ξ)=k$\omega _k(\xi )=k$ with a properly chosen k, the upper bound for λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ becomes as small as 32n+O(n−2)$\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of 2n+O(n−2)$\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper bound on λ̂n(ξ)$\widehat{\lambda }_n(\xi )$ for even values of n.

show abstract

“…It is known that for

n=2

\widehat{\lambda }_2(\xi )

can change between

\frac{1}{2}

and

\frac{\sqrt {5}-1}{2}

where both upper and lower bounds are sharp [11]. Recent developments about the spectrum of values

\widehat{\lambda }_2(\xi )

and

\widehat{\omega }_2(\xi )

can be found in [4, 9]. However, for

n\geqslant 3

and transcendental ξ, it is not even known if

\widehat{\lambda }_n(\xi )

can take any other value than

1/n

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Upper bounds for the uniform simultaneous Diophantine exponents

Badziahin

2022

Mathematika

View full text Add to dashboard Cite

show abstract

On uniform polynomial approximation

Poëls

2024

Journal De L’École Polytechnique — Mathématiques

View full text Add to dashboard Cite

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 ωn(ξ). Davenport and Schmidt's original 1969 inequality ωn(ξ) ⩽ 2n − 1 was improved recently, and the best upper bound known to date is 2n − 2 for each n ⩾ 10. In this paper, we develop new techniques leading us to the improved upper bound 2n − 1 3 n 1/3 + O(1). Résumé(Sur l'approximation polynomiale uniforme). -Soient n un entier strictement positif et ξ un nombre réel transcendant. Nous cherchons à borner supérieurement l'exposant uniforme d'approximation polynomiale ωn(ξ). Établie par Davenport et Schmidt en 1969, l'inégalité ωn(ξ) ⩽ 2n−1, a été améliorée pour la première fois récemment, et la meilleure borne supérieure connue à ce jour est 2n − 2 pour tout n ⩾ 10. Dans ce papier, nous développons de nouvelles techniques qui nous permettent d'obtenir la borne supérieure améliorée 2n − 1 3 n 1/3 + O(1). ContentsMathematical subject classification (2020). -11J13, 11J82.

show abstract

Exponents of Diophantine approximation in dimension 2 for a general class of numbers

Cited by 2 publications

References 27 publications

Upper bounds for the uniform simultaneous Diophantine exponents

Upper bounds for the uniform simultaneous Diophantine exponents

On uniform polynomial approximation

Contact Info

Product

Resources

About