Diophantine approximation of complex numbers

Author:

Nicola~M.R.~Oswald

2010 Mathematics Subject Classification:

11H06, 11J06, 11D09.

Keywords:

complex continued fractions, Diophantine approximation, geometry of numbers, lattices, Pell equation.

Abstract:

We give an upper bound for the approximation quality of diophantine approximations by quotients of lattice points in the complex plane. This upper bound depends on a certain lattice invariant. In particular, we generalize a method based on geometrical ideas of Hermann Minkowski and improved by Hilde Gintner. Subsequently we examine the spectrum arising from the infimum of the constants occuring in the upper bound and give a proof of the existence of infinitely many solutions of generalized Pell equations in the complex case.

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Vol. 10 (18), 2015