| Title:
|
Exponents for three-dimensional simultaneous Diophantine approximations (English) |
| Author:
|
Moshchevitin, Nikolay |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
127-137 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Theta = (\theta _1,\theta _2,\theta _3)\in \mathbb {R}^3$. Suppose that $1,\theta _1,\theta _2,\theta _3$ are linearly independent over $\mathbb {Z}$. For Diophantine exponents $$ \begin {aligned} \alpha (\Theta ) &= \sup \{\gamma >0\colon \limsup _{t\to +\infty } t^\gamma \psi _\Theta (t) <+\infty \},\\ \beta (\Theta ) &= \sup \{\gamma >0\colon \liminf _{t\to +\infty } t^\gamma \psi _\Theta (t)<+\infty \} \end {aligned} $$ we prove $$ \beta (\Theta ) \ge \frac {1}{2} \Bigg ( \frac {\alpha (\Theta )}{1-\alpha (\Theta )} +\sqrt {\Big (\frac {\alpha (\Theta )}{1-\alpha (\Theta )} \Big )^2 +\frac {4\alpha (\Theta )}{1-\alpha (\Theta )}} \Bigg ) \alpha (\Theta ). $$ (English) |
| Keyword:
|
Diophantine approximations |
| Keyword:
|
Diophantine exponents |
| Keyword:
|
Jarník's transference principle |
| MSC:
|
11J13 |
| idZBL:
|
Zbl 1249.11061 |
| idMR:
|
MR2899740 |
| DOI:
|
10.1007/s10587-012-0001-1 |
| . |
| Date available:
|
2012-03-05T07:17:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142046 |
| . |
| Reference:
|
[1] Jarník, V.: Contribution à la théorie des approximations diophantiennes linéaires et homogènes.Czech. Math. J. 4 (1954), 330-353 Russian, French summary. Zbl 0057.28303, MR 0072183 |
| Reference:
|
[2] Laurent, M.: Exponents of Diophantine approximations in dimension two.Can. J. Math. 61 (2009), 165-189. MR 2488454, 10.4153/CJM-2009-008-2 |
| Reference:
|
[3] Moshchevitin, N. G.: Contribution to Vojtěch Jarník.Preprint available at arXiv:0912.2442v3. MR 0095106 |
| Reference:
|
[4] Moshchevitin, N. G.: Khintchine's singular Diophantine systems and their applications.Russ. Math. Surv. 65 433-511 (2010), Translation from Uspekhi Mat. Nauk. 65 43-126 (2010). Zbl 1225.11094, MR 2682720, 10.1070/RM2010v065n03ABEH004680 |
| Reference:
|
[5] Schmidt, W. M.: On heights of algebraic subspaces and Diophantine approximations.Ann. Math. (2) 85 (1967), 430-472. Zbl 0152.03602, MR 0213301, 10.2307/1970352 |
| . |
