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

Ultrametric logarithm laws I

J. S. Athreya 1, , Anish Ghosh 2, and  Amritanshu Prasad 3,

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.
Keywords: Diophantine approximation, local fields., Shrinking target properties, logarithm laws.
Mathematics Subject Classification: 37A17, 37D40, 11K6.
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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