1. bookVolumen 17 (2022): Edición 1 (December 2022)
Detalles de la revista
License
Formato
Revista
eISSN
2309-5377
Primera edición
30 Dec 2013
Calendario de la edición
2 veces al año
Idiomas
Inglés
access type Acceso abierto

On Some Properties of Irrational Subspaces

Publicado en línea: 31 May 2022
Volumen & Edición: Volumen 17 (2022) - Edición 1 (December 2022)
Páginas: 89 - 104
Recibido: 23 Jul 2021
Aceptado: 26 Jan 2022
Detalles de la revista
License
Formato
Revista
eISSN
2309-5377
Primera edición
30 Dec 2013
Calendario de la edición
2 veces al año
Idiomas
Inglés
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.

Keywords

MSC 2010

[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

[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

[3] GROSHEV, A. V.: Un théorème sur les systèmes de formes linéaires, Dokl. Akad. Nauk SSSR 9 (1938), 151–152. Search in Google Scholar

[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

[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

[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

[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

[8] SCHMIDT, W. M.: Diophantine Approximation.In: Lecture Notes in Math. Vol. 785, Springer-Verlag, Berlin, 1980. Search in Google Scholar

[9] SCHMIDT, W. M.: Badly approximable systems of linear forms, J. Number Theory 1 (1969), no. 2, 139–154. Search in Google Scholar

Artículos recomendados de Trend MD

Planifique su conferencia remota con Sciendo