Spectral structure of digit sets of self-similar tiles on R1
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AbstractWe study the structure of the digit sets D for the integral selfsimilar tiles T (b, D) (we call such a D a tile digit set with respect to b). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial PD, i.e., the zeros of PD on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomials, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order. © 2013 American Mathematical Society.
All Author(s) ListLai C.-K., Lau K.-S., Rao H.
Journal nameTransactions of the American Mathematical Society
Volume Number365
Issue Number7
PublisherAmerican Mathematical Society
Place of PublicationUnited States
Pages3831 - 3850
LanguagesEnglish-United Kingdom
KeywordsBlocking, Cyclotomic polynomials, Kernel polynomials, Prime, Product-forms, Self-similar tiles, Spectra, Tile digit sets, Tree

Last updated on 2021-24-01 at 02:48