Fourier analysis of deformed model sets (Q2709121)

A regular model set of \(R^{n}\) is any subset \(X_{0}\subset R^{n}\) such that \(X_{0}+B=R^{n}\) for some open ball \(B\), and \(X_{0}\) is the canonical projection of some lattice \(L\)\ of rank \(m+n\) in \(R^{m+n}\) into \(R^{n}\). The authors define a deformed model set as \(X_{\varphi }=\{x+\varphi (\widetilde{x});x\in X_{0}\},\) where \(\varphi :R^{m}\rightarrow R^{n}\) is a continuous function with compact support and with small enough norm \(\sup |\varphi (y)|\) , \( \widetilde{x}\)\ belongs to the projection of \(L\) into \(R^{m}\). The Fourier-Bohr analysis of the model set \(X\) consists in looking for frequencies \(q\in R^{n}\) for which all the phases \(\exp \{-i2\pi qx\}\) , \( x\in X\), are coherent, or \(\lim_R{\rightarrow \infty} R^{-n}\sum_{x\in X,|x|<R}\exp \{-i2\pi qx\}\neq 0\). The corresponding approach to this problem for the regular model sets was developed by Y. Meyer and J. P. Schreiber, and the authors show that this approach is valid for deformed model sets. An example of application to the deformations of the Fibonacci chain is given.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00025].