| Title:
|
On a question of Schmidt and Summerer concerning $3$-systems (English) |
| Author:
|
Schleischitz, Johannes |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
28 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
253-262 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_{1},P_{2},P_{3})$ with the property $\overline {\varphi }_{3}=1$. In fact, our method generalizes to provide $n$-systems with $\overline {\varphi }_{n}=1$, for arbitrary $n\geq 3$. We visualize our constructions with graphics. We further present explicit examples of numbers $\xi _{1}, \ldots , \xi _{n-1}$ that induce the $n$-systems in question. (English) |
| Keyword:
|
parametric geometry of numbers |
| Keyword:
|
simultaneous approximation |
| MSC:
|
11H06 |
| MSC:
|
11J13 |
| idZBL:
|
Zbl 1475.11131 |
| idMR:
|
MR4197077 |
| . |
| Date available:
|
2021-03-03T08:55:16Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148705 |
| . |
| Reference:
|
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| Reference:
|
[2] Roy, D.: On Schmidt and Summerer parametric geometry of numbers.Annals of Mathematics, 2015, 739-786, JSTOR, MR 3418530, 10.4007/annals.2015.182.2.9 |
| Reference:
|
[3] Roy, D.: On the topology of Diophantine approximation spectra.Compositio Mathematica, 153, 7, 2017, 1512-1546, London Mathematical Society, MR 3705265, 10.1112/S0010437X17007126 |
| Reference:
|
[4] Schleischitz, J.: Diophantine approximation and special Liouville numbers.Communications in Mathematics, 21, 1, 2013, 39-76, MR 3067121 |
| Reference:
|
[5] Schleischitz, J.: On approximation constants for Liouville numbers.Glasnik matematički, 50, 2, 2015, 349-361, Hrvatsko matematičko društvo i PMF-Matematički odjel, Sveučilišta u Zagrebu, |
| Reference:
|
[6] Schmidt, W.M., Summerer, L.: Parametric geometry of numbers and applications.Acta Arithmetica, 140, 2009, 67-91, Instytut Matematyczny Polskiej Akademii Nauk, Zbl 1236.11060 |
| Reference:
|
[7] Schmidt, W.M., Summerer, L.: Diophantine approximation and parametric geometry of numbers.Monatshefte für Mathematik, 169, 1, 2013, 51-104, Springer, MR 3016519, 10.1007/s00605-012-0391-z |
| Reference:
|
[8] Schmidt, W.M., Summerer, L.: Simultaneous approximation to three numbers.Moscow Journal of Combinatorics and Number Theory, 3, 1, 2013, 84-107, MR 3284111 |
| Reference:
|
[9] Schmidt, W.M., Summerer, L.: Simultaneous approximation to two reals: bounds for the second successive minimum.Mathematika, 63, 3, 2017, 1136-1151, Wiley Online Library, MR 3731318, 10.1112/S0025579317000274 |
| . |
