[Author]
 Lodkin, A. A.

[Title]
Klein sail and Diophantine approximation of a vector 

[AMS Subj-class]
 11H06 Lattices and convex bodies; 
 11J70 Continued fractions and generalizations;
 52C07 Lattices and convex bodies in $n$ dimensions

[Keywords]
 Klein polyhedron, Klein sail, Diophantine approximation, golden number, 
 plastic number, Voronoi cell

[Abstract]
 In the works by Arnold and his followers, based on the ideas of Poincare 
and Klein, the Klein sails were built for some class of operators in $R^n$, 
and they were called multi-dimensional continued fractions. Another class 
of generalizations of continued fractions was related to sequences of 
rational bases approximating an irrational vector. We present a 
modification of the Klein's polyhedron and sail that is related directly  
to an irrational vector. We present a numerical characteristic of the sail 
related to a one-parameter group of transformations of a lattice and the 
induced deformation of the Voronoi cell and call it asymptotoc anisotropy.
This characteristic can probably give a geometric characterization of irrational 
vectors worst approximable by rational ones. In 2D, we give the candidate 
to such a vector which is related to the least Pisot number and may be 
considered a vector analog of the golden number.


[Comments]
 LaTeX, Russian, 4 pp.

[Contact e-mail]
 alodkin@gmail.com