DCDS
Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic
Sanghoon Kwon Seonhee Lim
Discrete & Continuous Dynamical Systems - A 2018, 38(1): 169-186 doi: 10.3934/dcds.2018008

For a local field K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H-orbit in SL(d, K)/SL(d, Z) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss.

We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class Cα.

keywords: Field of formal series effective equidistribution ergodic theorem Diophantine approximation
JMD
Volume entropy of hyperbolic buildings
François Ledrappier Seonhee Lim
Journal of Modern Dynamics 2010, 4(1): 139-165 doi: 10.3934/jmd.2010.4.139
We characterize the volume entropy of a regular building as the topological pressure of the geodesic flow on an apartment. We show that the entropy maximizing measure is not Liouville measure for any regular hyperbolic building. As a consequence, we obtain a strict lower bound on the volume entropy in terms of the branching numbers and the volume of the boundary polyhedrons.
keywords: building topological entropy geodesic flow. volume entropy volume growth
JMD
Asymptotic distribution of values of isotropic here quadratic forms at S-integral points
Jiyoung Han Seonhee Lim Keivan Mallahi-Karai
Journal of Modern Dynamics 2017, 11(1): 501-550 doi: 10.3934/jmd.2017020
We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places
$S$
over
${\mathbb{Q}}$
containing the Archimedean place and excluding the prime
$2$
, an irrational isotropic form
${\mathbf q}$
of rank
$n\geq 4$
on
${\mathbb{Q}}_S$
, a product of
$p$
-adic intervals
$\mathsf{I}_p$
, and a product
$\Omega$
of star-shaped sets. We show that unless
$n=4$
and
${\mathbf q}$
is split in at least one place, the number of
$S$
-integral vectors
$\mathbf v \in {\mathsf{T}} \Omega$
satisfying simultaneously
${\mathbf q}(\mathbf v) \in I_p$
for
$p \in S$
is asymptotically given by
$\begin{split}\lambda({\mathbf q}, \Omega) |\,\mathsf{I}\,| \cdot \| {\mathsf{T}} \|^{n-2}\end{split}$
as
${\mathsf{T}}$
goes to infinity, where
$|\,\mathsf{I}\,|$
is the product of Haar measures of the
$p$
-adic intervals
$I_p$
. The proof uses dynamics of unipotent flows on
$S$
-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an
$S$
-arithmetic variant of the
$ \alpha$
-function introduced in [10], and an
$S$
-arithemtic version of a theorem of Dani-Margulis [7].
keywords: Oppenheim conjecture homogeneous dynamics

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