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Journal of Modern Dynamics

2018 , Volume 12

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A quantitative Oppenheim theorem for generic ternary quadratic forms
Anish Ghosh and Dubi Kelmer
2018, 12: 1-8 doi: 10.3934/jmd.2018001 +[Abstract](1317) +[HTML](131) +[PDF](149.03KB)

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].

Values of random polynomials at integer points
Jayadev S. Athreya and Gregory A. Margulis
2018, 12: 9-16 doi: 10.3934/jmd.2018002 +[Abstract](84) +[HTML](45) +[PDF](140.36KB)

Using classical results of Rogers [12, Theorem 1] bounding the L2-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

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