Bounded Remainder Sets on a Double Covering of the Klein Bottle

Abstract

A shift S : K 2 −→ K 2 on the double covering of the Klein bottle K 2 = K 2 × {±1} is considered. This shift S generates a tiling K 2 = K 2 0 K 2 1 by two bounded remainder sets K 2 0 and K 2 1 with respect to the shift S. Two-sided bounds for the deviation functions of these sets are proved. Bibliography: 16 titles. 0. Introduction 0.1. Bounded remainder sets. Let, on a compact manifold M, a mapping M S −→ M be given and letOrbbe the orbit of a certain initial point x 0 ∈ M with respect to the mapping S. For an arbitrary set X ⊂ M, the distribution function is defined by the relationand the associated deviation function is defined as the differencewhere the constant a(X) ≥ 0 depends on the set X only. A set X is called a bounded remainder set or a BR-set with respect to the orbit (0.1) if there are some constants a(X) and c(X, x 0 ) such that the deviation function (0.3) satisfies the inequalityfor all i = 0, 1, 2, . . . . Toric bounded remainder sets.The structure of bounded remainder sets X on tori M = T D = R D /Z D of arbitrary dimensions D is understood sufficiently well [1][2][3][4][5][6][7]. Moreover, at present after a rather long pause [8][9][10][11] new ideas have appeared due to an extensive penetration of quasilattices and quasicrystals into different fields of mathematics and physics, and the interest in studying multidimensional toric bounded remainder sets has reappeared (see [6,7]). For instance, in [2-5] a general method for constructing such sets X was developed. The essence of this method is as follows: A classof bounded remainder sets P D ξ ξ , which are, in a sense, elementary, was distinguished; more complicated compound sets X are constructed from sets in the class Π.As to the sets P D ξ ξ , they are bounded remainder polyhedra, which are convex and nonconvex parallelohedra and admit continuous deformations [2,4]. In the small dimensions