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

100%

Li J.-L.

Studia Mathematica

2014

tom 220

nr 1

73-86

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.

Digit sets of integral self-affine tiles with prime determinant

100%

Li J.-L.

Studia Mathematica

2006

tom 177

nr 2

183-194

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 $D = M^{γ}D₀$, where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon, Lagarias and Wang. We then give several remarks and examples to illustrate some problems on digit sets.

Ograniczanie wyników

2 Studia Mathematica

2 Li J.-L.

1 2014

1 2006

Wyniki wyszukiwania

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

Digit sets of integral self-affine tiles with prime determinant