Klein sail and Diophantine approximation of a vector

A. A. Lodkin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 63-73

Résumé

In the papers by V. I. Arnold and his successors based upon the ideas of A. Poincaré and F. Klein, it was the Klein sail associated with an operator in $\mathbb R^n$ that they considered to play the role of a multidimensional continued fraction, and in these terms generalizations of the Lagrange theorem on continued fractions were formulated. A different approach to the generalization of the notion of continued fraction was based upon modifications of Euclid's algorithm for constructing, given an irrational vector, an approximating sequence of rational vectors. We suggest a modification of the Klein sail that is constructed directly from a vector, without any operator. A numeric characteristic of a Klein sail, its asymptotic anisotropy associated with a one-parameter transformation semigroup of the lattice that generates the sail, and of its Voronoï cell, is introduced. In terms of this anisotropy, we hope to give a geometric characterization of irrational vectors worst approximated by rational ones. In the three-dimensional space, we suggest a vector (related to the least Pisot–Vijayaraghavan number) that is a candidate for this role. This vector may be called an analog of the golden number, which is the worst approximated real number in the classical theory of Diophantine approximation.