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Spectrality of Self-Similar Tiles
We call a set with positive Lebesgue measure a {\it
spectral set} if admits an exponential orthonormal basis. It was
conjectured that is a spectral set if and only if is a tile (Fuglede's
conjecture). Despite the conjecture was proved to be false on ,
([T], [KM2]), it still poses challenging questions with additional
assumptions. In this paper, our additional assumption is self-similarity. We
study the spectral properties for the class of self-similar tiles in
that has a product structure on the associated digit sets. We
show that any strict product-form tiles and the associated modulo product-form
tiles are spectral sets. As for the converse question, we give a pilot study
for the self-similar set generated by arbitrary digit sets with four
elements. We investigate the zeros of its Fourier transform due to the
orthogonality, and verify Fuglede's conjecture for this special case.Comment: 22page
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