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Trigonometric inequalities for Fibonacci chains

Anna Chiara Lai, Paola Loreti

TL;DR

This work addresses nonharmonic Fourier series with frequencies drawn from $m$-bonacci chains $\Lambda_m$ generated by Rauzy substitutions. By connecting digit frequencies of the $m$-bonacci words to the chain's upper density $D^+(\Lambda_m)$ and applying Beurling-type inequalities, it establishes explicit Ingham-type trigonometric inequalities: for intervals with length above $D^+(\Lambda_m)$, one obtains norm equivalences for Fourier sums with frequencies in $\Lambda_m$. The paper provides a precise formula $D^+(\Lambda_m)=\frac{\rho_m^{2m}}{1+\rho_m^2+\cdots+\rho_m^{2(m-1)}}$, demonstrates that $D^+(\Lambda_m)\uparrow 3$ as $m\to\infty$, and supplies gap conditions for the Fibonacci ($m=2$) and Tribonacci ($m=3$) chains by exploiting detailed digit-weight analyses of the corresponding words. These results fuse symbolic dynamics and nonharmonic analysis, with potential implications for spectrography and diffraction patterns in one-dimensional quasicrystals.

Abstract

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.

Trigonometric inequalities for Fibonacci chains

TL;DR

This work addresses nonharmonic Fourier series with frequencies drawn from -bonacci chains generated by Rauzy substitutions. By connecting digit frequencies of the -bonacci words to the chain's upper density and applying Beurling-type inequalities, it establishes explicit Ingham-type trigonometric inequalities: for intervals with length above , one obtains norm equivalences for Fourier sums with frequencies in . The paper provides a precise formula , demonstrates that as , and supplies gap conditions for the Fibonacci () and Tribonacci () chains by exploiting detailed digit-weight analyses of the corresponding words. These results fuse symbolic dynamics and nonharmonic analysis, with potential implications for spectrography and diffraction patterns in one-dimensional quasicrystals.

Abstract

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 -bonacci chains. The result is achieved by characterizing the upper density of the -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.

Paper Structure

This paper contains 6 sections, 8 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Rauzy substitutions
  3. Ingham-type inequalities for the $m$-bonacci chains
  4. Gap conditions in the Fibonacci chain and in the Tribonacci chain
  5. Digit weights for the Fibonacci and Tribonacci words
  6. Some gap conditions for $m=2,3$.

Key Result

Proposition 1

For all $m\geq 2$, $\sigma_m$ is a Pisot substitution. In particular, the Perron-Frobenious eigevanlue $\rho_m$ of $\sigma_m$ is the Pisot number whose minimal polynomial $\lambda^m-\lambda^{m-1}-\cdots-\lambda-1$. Moreover the vector is a left eigenvector associated to $\rho_m$ whose $\ell_0$ norm is equal to $1$.

Figures (1)

  • Figure 1: An approximation of $D^+(\Lambda_2)$ via $\mathbf n(r)/r$ with $r\in[250,500]$.

Theorems & Definitions (22)

  • Definition 1: Uniform frequency
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2: Rauzy words and their frequencies
  • Example 1: Fibonacci word
  • Example 2: Tribonacci word
  • Remark 1
  • Definition 3: $m$-bonacci chains
  • Definition 4: Upper density
  • ...and 12 more