Trigonometric inequalities for Fibonacci chains

In this work, we consider m-bonacci chains, unidimensional quasicrystals obtained by general classes of Rauzy substitutions. Motivated by applications in spectrography and diffraction patterns of some quasicrystals, we pose the problem of establishing Ingham type trigonometric inequalities when the frequencies belong to $m$-bonacci chains. The result is achieved by characterizing the upper density of the $m$-bonacci chains. Tools from symbolic dynamics and combinatorics on words are used. Explicit gap conditions for the particular cases of Fibonacci chains and Tribonacci chains complete the paper.