<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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 <italic>ξ</italic> from two-dimensional badly approximable completely irrational subspace of ℝ<italic><sup>d</sup></italic> one has <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0002_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mi>ω</mi><mo>⌢</mo></mover><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><msqrt><mrow><mn>5</mn><mo>-</mo><mn>1</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></math>
<tex-math>\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \omega } \left( \xi  \right) \le {{\sqrt {5 - 1} } \over 2}</tex-math>
</alternatives>
</inline-formula>. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.</p>
</abstract>

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 ℝd one has 


ω⌢(ξ)≤5-12
\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.

On Some Properties of Irrational Subspaces

Department of Number Theory, Faculty of Mechanics and Mathematics

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On Some Properties of Irrational Subspaces"}

In this paper, we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly...

On Some Properties of Irrational Subspaces

Publié en ligne: 31 mai 2022

Pages: 89 - 104

Reçu: 23 juil. 2021

Accepté: 26 janv. 2022

DOI: https://doi.org/10.2478/udt-2022-0002

Mots clés
badly approximable matrices, completely irrational subspaces, ()-games

© 2022 Vasiliy Neckrasov, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On Some Properties of Irrational Subspaces

Publié en ligne: 31 mai 2022

Pages: 89 - 104

Reçu: 23 juil. 2021

Accepté: 26 janv. 2022

DOI: https://doi.org/10.2478/udt-2022-0002

Mots clésbadly approximable matrices, completely irrational subspaces, ()-games

© 2022 Vasiliy Neckrasov, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
badly approximable matrices, completely irrational subspaces, ()-games