Optimal Diophantine Exponents for $\mathrm{SL}(n)$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910376394752000 |
|---|---|
| author | Jana, Subhajit Kamber, Amitay |
| author_facet | Jana, Subhajit Kamber, Amitay |
| contents | The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05106 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Optimal Diophantine Exponents for $\mathrm{SL}(n)$ Jana, Subhajit Kamber, Amitay Number Theory Dynamical Systems 11F70, 11J13 The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the $\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$, substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal}, i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the \emph{temperedness} of the underlying representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local $L^2$-norms of the Eisenstein series. |
| title | Optimal Diophantine Exponents for $\mathrm{SL}(n)$ |
| topic | Number Theory Dynamical Systems 11F70, 11J13 |
| url | https://arxiv.org/abs/2211.05106 |