Self-similarity of a family of binary quasiperiodic sequences associated with quadratic irrationals

Abstract

Binary quasiperiodic (QP) sequences of 1 and 0 generated by a projection method, f k = [ (k + 1) (1 + ω) + θ 0] − [ k (1 + ω) + θ 0] with θ 0 = 0, preserve self-similarity if and only if ω is a quadratic irrational number (see, e.g. the paper by Odagaki and Kaneko [J. Phys. A 27 (1994) 1683]). Here [ x] is the integer part of x, k are integers, and the quantity θ 0 denotes the shift in the position of the strip in the projection method. In this paper, the necessary and sufficient conditions for the QP sequences f k to be self-similar with a non-vanishing θ 0 are established.