Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
convex geometry; lattices; Liouville numbers; successive minima
convex geometry; lattices; Liouville numbers; successive minima
Summary:
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},\ldots ,\zeta_{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\geq 2$) of $\zeta_{1},\zeta_{2},\ldots ,\zeta_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $\zeta_{1},\zeta_{2},\ldots ,\zeta_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.
References:
[1] Gruber, P.M., Lekkerkerker, C.G.: Geometry of numbers. 1987, North-Holland Verlag, MR 0893813 | Zbl 0611.10017
[2] Jarník, V.: Contribution to the theory of linear homogeneous diophantine approximation. Czechoslovak Math. J., 4, 79, 1954, 330-353, MR 0072183
[3] Moshchevitin, N.G.: Proof of W.M. Schmidt's conjecture concerning successive minima of a lattice. J. London Math. Soc., 2012, doi: 10.1112/jlms/jdr076, 12 Mar 2012. MR 2959298
[4] Roy, D.: Diophantine approximation in small degree. 2004, Number theory 269--285, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI. MR 2076601 | Zbl 1077.11051
[5] Schmidt, W.M., Summerer, L.: Parametric geometry of numbers and applications. Acta Arithm., 140, 1, 2009, DOI 10.4064/aa140-1-5 | MR 2557854 | Zbl 1236.11060
[6] Schmidt, W.M., Summerer, L.: Diophantine approximation and parametric geometry of numbers. to appear in Monatshefte für Mathematik. MR 3016519 | Zbl 1264.11056
[7] Waldschmidt, M.: Report on some recent advances in Diophantine approximation. 2009,
Search
Partner of
