# American Institute of Mathematical Sciences

2009, 2(2): 337-348. doi: 10.3934/dcdss.2009.2.337

## Ultrametric logarithm laws I

 1 Department of Mathematics, Princeton University, Fine Hall, Washington Road Princeton NJ 08544-1000, United States 2 Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200 Austin, Texas 78712, United States 3 The Institute of Mathematical Sciences, CIT campus, Taramani, Chennai 600 113, India

Received  May 2008 Revised  November 2008 Published  April 2009

We announce ultrametric analogues of the results of Kleinbock-Margulis for shrinking target properties of semisimple group actions on symmetric spaces. The main applications are $S$-arithmetic Diophantine approximation results and logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on trees.
Citation: J. S. Athreya, Anish Ghosh, Amritanshu Prasad. Ultrametric logarithm laws I. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 337-348. doi: 10.3934/dcdss.2009.2.337
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