| Title:
|
Diophantine approximation and special Liouville numbers (English) |
| Author:
|
Schleischitz, Johannes |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
21 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
39-76 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
convex geometry |
| Keyword:
|
lattices |
| Keyword:
|
Liouville numbers |
| Keyword:
|
successive minima |
| MSC:
|
11H06 |
| MSC:
|
11J13 |
| MSC:
|
11J81 |
| idZBL:
|
Zbl 06202724 |
| idMR:
|
MR3067121 |
| . |
| Date available:
|
2013-07-18T15:26:29Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143348 |
| . |
| Reference:
|
[1] Gruber, P.M., Lekkerkerker, C.G.: Geometry of numbers.1987, North-Holland Verlag, Zbl 0611.10017, MR 0893813 |
| Reference:
|
[2] Jarník, V.: Contribution to the theory of linear homogeneous diophantine approximation.Czechoslovak Math. J., 4, 79, 1954, 330-353, MR 0072183 |
| Reference:
|
[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 |
| Reference:
|
[4] Roy, D.: Diophantine approximation in small degree.2004, Number theory 269--285, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI. Zbl 1077.11051, MR 2076601 |
| Reference:
|
[5] Schmidt, W.M., Summerer, L.: Parametric geometry of numbers and applications.Acta Arithm., 140, 1, 2009, Zbl 1236.11060, MR 2557854, 10.4064/aa140-1-5 |
| Reference:
|
[6] Schmidt, W.M., Summerer, L.: Diophantine approximation and parametric geometry of numbers.to appear in Monatshefte für Mathematik. Zbl 1264.11056, MR 3016519 |
| Reference:
|
[7] Waldschmidt, M.: Report on some recent advances in Diophantine approximation.2009, |
| . |
