Classification of tile digit sets as product-forms
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AbstractLet A be an expanding matrix on R-s with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set D subset of Z(s) so that the integral self-affine set T(A, D) is a translational tile on R-s. In our previous paper, we classified such tile digit sets D subset of Z by expressing the mask polynomial P-D as a product of cyclotomic polynomials. In this paper, we first show that a tile digit set in Z(s) must be an integer tile (i. e., D circle plus L = Z(s) for some discrete set L). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on R-1 together with our previous results to characterize explicitly all tile digit setsD. Z with A = p(alpha)q (p, q distinct primes) as modulo product-form of some order, an advance of the previously known results for A = p(alpha) and pq.
All Author(s) ListLai CK, Lau KS, Rao H
Journal nameTransactions of the American Mathematical Society
Detailed descriptionTo ORKTS: Article electronically published on April 15, 2016
Year2017
Month1
Volume Number369
Issue Number1
PublisherAMER MATHEMATICAL SOC
Pages623 - 644
ISSN0002-9947
eISSN1088-6850
LanguagesEnglish-United Kingdom
KeywordsBlocking; cyclotomic polynomials; integer tiles; kernel polynomials; prime; product-forms; self-affine tiles; spectra; tile digit sets; tree
Web of Science Subject CategoriesMathematics

Last updated on 2021-16-01 at 01:22