On Some Properties of Irrational Subspaces

Vasiliy Neckrasov

doi:10.2478/udt-2022-0002

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 ℝ^d one has $\overset{⌢}{ω} (ξ) \leq \frac{\sqrt{5 - 1}}{2}$ \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.

References

[1] BRODERICK, R.—FISHMAN, L.—KLEINBOCK, D.—REICH, A.—WEISS, B.: The set of badly approximable vectors is strongly C1 incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319–339.
Search in Google Scholar Back to article
[2] BRODERICK, R.—FISHMAN, L.—SIMMONS, D.: Badly approximable systems of affine forms and incompressibility on fractals, J. Number Theory 133 (2013), no. 7, 2186–2205.
Search in Google Scholar Back to article
[3] GROSHEV, A. V.: Un théorème sur les systèmes de formes linéaires, Dokl. Akad. Nauk SSSR 9 (1938), 151–152.
Search in Google Scholar Back to article
[4] KLEINBOCK, D.—MOSHCHEVITIN, N. G.—WEISS, B.: singular vectors on manifolds and fractals,Israel J. Math. 245 (2021), 589–613.10.1007/s11856-021-2220-3
Search in Google Scholar Back to article
[5] MARNAT, A.—MOSHCHEVITIN, N. G.: An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation,Mathematika 66 (2020), no. 3, 818–854.
Search in Google Scholar Back to article
[6] MOSHCHEVITIN, N. G.: Khintchine’s singular Diophantine systems and their applications, Russian Math. Surveys 65 (2010), no. 3, 433–511.
Search in Google Scholar Back to article
[7] SCHLEISCHITZ, J.: Applications of Siegel’s Lemma to a system of linear forms and its minimal points, preprint, available at arXiv:1904.06121.
Search in Google Scholar Back to article
[8] SCHMIDT, W. M.: Diophantine Approximation.In: Lecture Notes in Math. Vol. 785, Springer-Verlag, Berlin, 1980.
Search in Google Scholar Back to article
[9] SCHMIDT, W. M.: Badly approximable systems of linear forms, J. Number Theory 1 (1969), no. 2, 139–154.
Search in Google Scholar Back to article

On Some Properties of Irrational Subspaces

Abstract

Paradigm

My account