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Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket. (English) Zbl 1166.28004

Let \(M\) be an expanding \(n\)-by-\(n\) integer matrix \((|\lambda_i|> 1\) for all eigenvalues \(\lambda_i\) of \(M\)) and let \(D\subseteq\mathbb{Z}^n\) be a finite set of cardinality \(|D|\).
Given an iterated function system (IFS) \(\{\phi_d(x)= M^{-1}(x+ d)\}_{d\in D}\), there is a unique probability measure \(\mu\) satisfying
\[ \mu={1\over|D|} \sum_{d\in D} \mu\circ\phi^{- 1}_d, \]
supported on a set \(T(M, D)\), called a self-affine measure. Such a measure \(\mu\) on \(\mathbb{R}^n\) with compact support is called a spectral measure if there exists a discrete set \(\Lambda\subseteq\mathbb{R}^n\) such that \(E_\Lambda= \{e^{2\pi i(\lambda, x)},\lambda\in \Lambda\}\) forms an orthogonal basis for \(L^2(\mu)\). \(\Lambda\) is called the spectrum of \(\mu\). The spectral self-affine measure problem consists in determining conditions under which \(\mu= \mu_{M,D}\) is a spectral measure.
This paper is concerned with the related problem concerning non-spectral sets. It is shown that the self-affine measure \(\mu_{M,D}\) corresponding to
\[ M= \left[\begin{matrix} p & 0 & m\\ 0 & p & 0\\ 0 & 0 & p\end{matrix}\right],\quad D= \left\{\begin{pmatrix} 0\\ 0\\ 0\end{pmatrix}, \begin{pmatrix} 1\\ 0\\ 0\end{pmatrix}, \begin{pmatrix} 0\\ 1\\ 0\end{pmatrix}, \begin{pmatrix} 0\\ 0\\ 1\end{pmatrix}\right\} \]
(\(p\) odd), is supported on a generalized Siepiński gasket and that there exists at most seven mutually orthogonal exponential functions in \(L^2(\mu_{M,D})\). Other examples of this type are considered. This work improves results of D. E. Dutkay and P. E. T. Jorgensen [J. Funct. Anal. 247, No. 1, 110–137 (2007; Zbl 1128.42013)] (see also J.-L. Li [J. Approx. Theory 153, No. 2, 161–169 (2008; Zbl 1214.42046)]).

MSC:

28A80 Fractals
42C99 Nontrigonometric harmonic analysis
37E99 Low-dimensional dynamical systems
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