Journal of Modern Dynamics (JMD)

Approximation of points in the plane by generic lattice orbits

Pages: 143 - 153, Volume 11, 2017      doi:10.3934/jmd.2017007

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Dubi Kelmer - Department of Mathematics, Maloney Hall, Boston College, Chestnut Hill, MA 02467-3806, United States (email)

Abstract: We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma < {SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives bounds that are uniform for almost all orbits.

Keywords:  Diophantine approximation, lattice action, shrinking targets.
Mathematics Subject Classification:  Primary: 11J20; Secondary: 37A17.

Received: June 2016;      Revised: December 2016;      Available Online: February 2017.