A quantitative Oppenheim theorem for generic ternary quadratic forms
Anish Ghosh Dubi Kelmer
Journal of Modern Dynamics 2018, 12(1): 1-8 doi: 10.3934/jmd.2018001

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain [3].

keywords: Values of quadratic forms Diophantine approximation flows on homogeneous spaces
Ultrametric logarithm laws I
J. S. Athreya Anish Ghosh Amritanshu Prasad
Discrete & Continuous Dynamical Systems - S 2009, 2(2): 337-348 doi: 10.3934/dcdss.2009.2.337
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

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