Kolakoski-(3, 1) Is a (Deformed) Model Set

Abstract

Abstract. Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding biinfinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.

Kolakoski sequences – an example of aperiodic order

“…The eigenvectors of their substitution matrices M yield a lattice C, which is two-dimensional in the case of Fibonacci, but threedimensional in the case of Kol (3,1). The orthogonal projection of lattice points in a certain strip through this lattice result in a geometrical representation of the sequence in question, e.g., in Kol(3, 1) the letter A of (8) is represented as interval of length ' A ¼ a 2 À a % 2:66, B as one of length ' B ¼ a % 2:21 and C as one of length ' C ¼ 1 (note that the lattice points are projected on the left end points of these intervals), see [5], where also the explicit construction of the lattice C is given and where it is proved that Kol(3, 1) really is a model set. An important role, especially for diffraction is played by the cross-section of the projecting strip, the window X.…”

Deformed Penrose tilings

“…More recently Baake and co-workers have been involved in the development of mathematical models called deformed model sets [10][11][12][13] and this work has provided some rigorous proofs for such deformed tilings. In the present context a model set essentially means that it can be produced by the cut-and-project method from a higher-dimensional crystal.…”

Some Formulas for the Generalized Kolakoski Sequence <i>Kol</i>(<i>a, b</i>)

“…∆ The case where a and b are odd, has been explored by Baake et al [1] who found a connection between the generalized Kolakoski sequence and some deformed model sets. They used Perron-fronebius Theorem and found that the frequency of '3' in ( ) 1,3 Kol is " 0.60 ≈ ".…”

“…They used Perron-fronebius Theorem and found that the frequency of '3' in ( ) 1,3 Kol is " 0.60 ≈ ".…”