I am interested in finding an intuitive way to show that the Fibonacci chain is quasiperiodic (and not simply random). Or put differently, how can I tell from just looking at a given chain whether or not it is quasiperiodic?
Let us consider the construction of the Fibonacci chain. We act with the following substitution rule \begin{eqnarray} S &\rightarrow& L\\ \nonumber L &\rightarrow& LS \end{eqnarray} many times on a starting letter, e.g. $L$, in order to construct a long word (=chain). The second line means the replacement of one letter $L$ by two letters ($L$ and $S$). This leads to the aperiodic word (Fibonacci chain) of the following form \begin{eqnarray} L \rightarrow LS \rightarrow LSL \rightarrow LSLLS \rightarrow LSLLSLSL \rightarrow LSLLSLSLLSLLS \rightarrow \dots \end{eqnarray}
I can prove that this word is indeed aperiodic (see also this question). Also, it is clear that it is not random since we used a rule (not a random number generator) to construct the sequence.
Now I am going convert this word into a (fictious) physical atomic structure ('quasicrystal'). The atoms sit in between the long ($L$) and short ($S$) segments.
How can I easily show that it is quasiperiodic?
I know that quasicrystals are defined by having sharp diffraction peaks. So I can numerically (or analytically) calculate the diffraction pattern from the Fibonacci chain. In this case I calculated the Fourier transform of a Fibonacci chain with 21 segments, namely \begin{equation} L, S, L, L, S, L, S, L, L, S, L, L, S, L, S, L, L, S, L, S, L \end{equation} with $S = 1$, $L = $ golden mean and I see it has sharp peaks:
Then I compare it to that of a pseudorandom chain \begin{equation} S, L, S, S, S, S, L, S, S, S, L, L, S, L, S, L, S, L, S, S, S \end{equation} which is generated randomly.
Ok, one could argue these peaks are not as 'sharp' as in the QC case but this depends a lot on the definition of 'sharp' (or, rather, 'discrete'). Also, I know that my example is probably not very well-defined because on a 21-long chain the difference between random and quasiperiodic might be hard to pin down. However, it would be nice to find a more elegant argument without me having to zoom into the diffraction pattern of extremely long Fibonacci/random chains (which cost a lot of numerical effort) to show the difference.
For example, could there be an argument along the lines of 'almost periodic' which is nicely illustrated here, section 5.2?
I would like to keep self-similarity out of the discussion here because, in principle, any Fibonacci sequence (not just the typical self-similar one I took with $S = 1$ and $L$ = golden mean) should be quasiperiodic.
