## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

DCDS

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 *C _{α}*.

JMD

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.

JMD

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10 ]. Suppose we are given a finite set of places

over

containing the Archimedean place and excluding the prime

, an irrational isotropic form

of rank

on

, a product of

-adic intervals

, and a product

of star-shaped sets. We show that unless

and

is split in at least one place, the number of

-integral vectors

satisfying simultaneously

for

is asymptotically given by

$S$ |

${\mathbb{Q}}$ |

$2$ |

${\mathbf q}$ |

$n\geq 4$ |

${\mathbb{Q}}_S$ |

$p$ |

$\mathsf{I}_p$ |

$\Omega$ |

$n=4$ |

${\mathbf q}$ |

$S$ |

$\mathbf v \in {\mathsf{T}} \Omega$ |

${\mathbf q}(\mathbf v) \in I_p$ |

$p \in S$ |

$\begin{split}\lambda({\mathbf q}, \Omega) |\,\mathsf{I}\,| \cdot \| {\mathsf{T}} \|^{n-2}\end{split}$ |

as

goes to infinity, where

is the product of Haar measures of the

-adic intervals

. The proof uses dynamics of unipotent flows on

-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

-arithmetic variant of the

-function introduced in [10 ], and an

-arithemtic version of a theorem of Dani-Margulis [7 ].

${\mathsf{T}}$ |

$|\,\mathsf{I}\,|$ |

$p$ |

$I_p$ |

$S$ |

$S$ |

$ \alpha$ |

$S$ |

## Year of publication

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