Exact approximation order and well-distributed sets

Bandi, Prasuna; Ghosh, Anish; Nandi, Debanjan

doi:10.1016/j.aim.2023.108871

Advances in Mathematics

2023

DOI: 10.1016/j.aim.2023.108871

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Exact approximation order and well-distributed sets

Prasuna Bandi

Anish Ghosh

Debanjan Nandi

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2024

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Cited by 3 publications

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“…denotes the lower order at infinity of the function Ψ −1 . The high dimensional case and the case for approximation in the complex number field were considered by Bandi and de Saxcé [2] and He and Xiong [11], respectively. A general framework for exact approximation was introduced by Bandi, Ghosh, and Nandi [1] by introducing the notion of a well-distributed set.…”

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confidence: 99%

Exact Diophantine approximation of real numbers by $ \beta $-expansions

Zhang,

Zhong

2024

DCDS

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confidence: 99%

Exact Diophantine approximation of real numbers by $ \beta $-expansions

Zhang,

Zhong

2024

DCDS

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A remark on the set of exactly approximable vectors in the simultaneous case

Fregoli

2024

Proc. Amer. Math. Soc.

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We compute the Hausdorff dimension of the set of ψ \psi -exactly approximable vectors, in the simultaneous case, in dimension strictly larger than 2 2 and for approximating functions ψ \psi with order at infinity less than or equal to − 2 -2 . Our method relies on the analogous result in dimension 1 1 , proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.

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The Dimension of the Set of ψ-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani

Koivusalo,

Levesley,

Ward

et al. 2024

International Mathematics Research Notices

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Let $\psi :{\mathbb{N}} \to [0,\infty )$, $\psi (q)=q^{-(1+\tau )}$ and let $\psi $-badly approximable points be those vectors in ${\mathbb{R}}^{d}$ that are $\psi $-well approximable, but not $c\psi $-well approximable for arbitrarily small constants $c>0$. We establish that the $\psi $-badly approximable points have the Hausdorff dimension of the $\psi $-well approximable points, the dimension taking the value $(d+1)/(\tau +1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large $\liminf $ set and combine this with ideas inspired by the proof of the MTP to find a large $\limsup $ subset of the $\liminf $ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.

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Exact approximation order and well-distributed sets

Cited by 3 publications

References 16 publications

Exact Diophantine approximation of real numbers by $ \beta $-expansions

Exact Diophantine approximation of real numbers by $ \beta $-expansions

A remark on the set of exactly approximable vectors in the simultaneous case

The Dimension of the Set of ψ-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani

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