Diophantine exponents for standard linear actions of ${\rm SL}_2$ over discrete rings in $\mathbb {C}$

Abstract

We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of SL 2 (O K ) on the punctured complex plane C 2 \ {0}, where K is a number field whose ring of integers O K is discrete and within a unit distance of any complex number. The results are similar to those of Laurent and Nogueira for SL 2 (Z) action on R 2 \ {0} albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

“…The argument here is same as the one used in [10,16] except that we get tighter bounds owing to the ultrametric inequality. Now if k is chosen so as to have | Q k | ≤ | γ | < | Q k+1 |, we immediately get µ(x, 0) ≤ 1.…”

“…Laurent and Nogueira [10] confined their investigations to the standard linear action of the lattice SL(2, Z) on the punctured plane R 2 \ {0}. In a previous work [16], the second-named author extended their approach and showed similar results for a few lattices inside SL(2, C) acting linearly on C 2 \ {0}. The last two approaches involve making use of some continued fraction algorithm to construct certain convergent matrices belonging to the relevant lattice.…”

On Kronecker's density theorem, primitive points and orbits of matrices

“…Singhal [15] has partly extended the results of Theorem 3 to the linear action of the group SL 2 (O K ) on C 2 , where O K stands for the ring of integers of an imaginary quadratic field K for which a convenient theory of continued fractions is available.…”

Approximation of points in the plane by generic lattice orbits

“…One can also consider the same problem for the action of lattices Γ ⊆ SL 2 (C) acting on C 2 . There have been a few results in this case: for Γ = SL 2 (O) with O = Z[i] the ring of Gaussian integers recent results of Singhal [Sin15] imply that…”