SELF-SIMILARITY OF A FAMILY OF BINARY QUASI-PERIODIC SEQUENCES ASSOCIATED WITH QUADRATIC IRRATIONALS

Binary quasiperiodic (QP) sequences of 1 and 0 generated by a projecti on method, =[(k+1)/(1+omega)+theta(0)]-[k/(1+omega)+theta(0)] with the ta(0)=0, preserve self-similarity if and only if omega 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 theta(0) denotes the shift in the position of the s trip in the projection method. In this paper, the necessary and suffic ient conditions for the QP sequences f(k) to be self-similar with a no n-vanishing theta(0) are established.