A continued fraction of a inhomogeneous linear form

Abstract:

Let α,β be real numbers 0≤α<1,0≤β<1. They define at the plane (y,z)∈R^{2} the inhomogeneous linear form L_{α,β}(y,z)=-β+αy+z. We propose the algorithm of an expansion of this linear form into the 'inhomogeneous continued fraction' L_{α,β}~[0;b_{1},b_{2},...]mod[0;a_{1},a_{2},...]. Inhomogeneous continued fraction generalize the classic regular continued fraction: for β=0 every b_{n}=0 and we get the continued fraction expansion of the number α: L_{α,0}~[0]mod[0;a_{1},a_{2},...]. Some properties of inhomogeneous continued fractions are proved.

Keywords:

continued fractions, inhomogeneous Diophantine approximations, Euclid algorithm

Publication language:russian,
pages:15

Research direction:

Mathematical modelling in actual problems of science and technics