Consider a sequence {an}$\{an\}$ consisting of fractional parts of the numbers an$an$ for natural numbers n$n$ where a$a$ is an irrational number. Its nth$n$th value is in the interval (x;y)$(x;y)$ for numbersn = n_1,n_2,\dotsnumbers $n = n_1,n_2,\dots$. Consider the sequence of differences n_2-n_1,n_3-n_2,\dots It$n_2-n_1,n_3-n_2,\dots$. It is a quasiperiodic sequence. Can one obtain it by recursive transformations of a string?
For example, the sequence of numbers n$n$ such that {\sqrt{2}n}$\{n\sqrt2\}$ is inside the interval (0,0.2)$(0,0.2)$ is transformed into a sequence consisting of the lines "5 5 7 5 5 2", "5 5 7 5 7 5 5 2" and "5 5 7 5 7 5 5 7 5 7 5 5 2". If we denote them as 6, 8 and 13 respectively, we obtain the blocks of the form "6 8 6 8 6 6 8 6 8 6 13", "6 8 6 8 6 6 8 6 8 6 6 8 6 8 6 8 6 6 8 6 8 6 6 8 6 8 6 13" and "6 8 6 8 6 6 8 6 8 6 6 8 6 8 6 8 6 6 8 6 8 6 6 8 6 8 6 8 6 6 8 6 8 6 6 8 6 8 6 13". Has it been shown that one can continue this process indefinitely and always use at most 3 different types of blocks?
