Singular vectors on manifolds and fractals

Kleinbock, Dmitry; Moshchevitin, Nikolay; Weiss, Barak

doi:10.1007/s11856-021-2220-3

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…ÈÖÓÔÓ× Ø ÓÒ 3.1º Let L be an n − dimensional badly approximable linear subspace of R d . Then As it was shown in [4], inequality (10) follows from…”

Section: Some Upper Bounds For the Exponentsmentioning

confidence: 71%

“…It is well-known that for irrational ξ we have 1 d−1 ≤ ω(ξ) ≤ 1 and that ω(ξ) ≥ ω(ξ). The following Proposition 3.1 was proved in [4]. Here we should mention that the notation here differs from [4].…”

Section: Some Upper Bounds For the Exponentsmentioning

confidence: 95%

“…Here we should mention that the notation here differs from [4]. In particular, our constants w d,n and W n,d can be obtained from those in [4] by substitution…”

Section: Some Upper Bounds For the Exponentsmentioning

confidence: 99%

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On Some Properties of Irrational Subspaces

Neckrasov

2022

Uniform Distribution Theory

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In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector ξ from two-dimensional badly approximable completely irrational subspace of ℝ d one has ω ⌢ ( ξ ) ≤ 5 - 1 2 \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi \right) \le {{\sqrt {5 - 1} } \over 2} . Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.

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“…ÈÖÓÔÓ× Ø ÓÒ 3.1º Let L be an n − dimensional badly approximable linear subspace of R d . Then As it was shown in [4], inequality (10) follows from…”

Section: Some Upper Bounds For the Exponentsmentioning

confidence: 71%

Section: Some Upper Bounds For the Exponentsmentioning

confidence: 95%

See 1 more Smart Citation

On Some Properties of Irrational Subspaces

Neckrasov

2022

Uniform Distribution Theory

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Dirichlet is not just bad and singular in many rational IFS fractals

Schleischitz

2023

The Quarterly Journal of Mathematics

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For $m\ge 2$, consider K the m-fold Cartesian product of the limit set of an iterated function system (IFS) of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and K does not degenerate to singleton, we construct vectors in K that lie within the ‘folklore set’ as defined by Beresnevich et al., meaning that they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within K; however, we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.

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On Angles between Linear Subspaces in $$\mathbb R^4$$ and the Singularity

Chebotarenko

2024

Math Notes

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Singular vectors on manifolds and fractals

Cited by 6 publications

References 26 publications

On Some Properties of Irrational Subspaces

On Some Properties of Irrational Subspaces

Dirichlet is not just bad and singular in many rational IFS fractals

On Angles between Linear Subspaces in $$\mathbb R^4$$ and the Singularity

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