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Digit sets of integral self-affine tiles with prime determinant

Jian-Lin Li — 2006

Studia Mathematica

Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ϕ d ( x ) = M - 1 ( x + d ) d ∈ D has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that D = M γ D ₀ , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon,...

Sufficient conditions for the spectrality of self-affine measures with prime determinant

Jian-Lin Li — 2014

Studia Mathematica

Let μ M , D be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of μ M , D when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for μ M , D to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.

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