<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

Published Online: May 31, 2022

Page range: 89 - 104

Received: Jul 23, 2021

Accepted: Jan 26, 2022

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

Keywords
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

Published Online: May 31, 2022

Page range: 89 - 104

Received: Jul 23, 2021

Accepted: Jan 26, 2022

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

Keywordsbadly 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.

Keywords
badly approximable matrices, completely irrational subspaces, ()-games