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A Hajós type result on factoring finite abelian groups by subsets. II

Keresztély Corrádi, Sándor Szabó (2010)

Commentationes Mathematicae Universitatis Carolinae

It is proved that if a finite abelian group is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be periodic. This result generalizes Hajós's factorization theorem.

Complex Hadamard matrices and the spectral set conjecture.

Mihail N. Kolountzakis, Máté Matolcsi (2006)

Collectanea Mathematica

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral ⇒ tile" of the Spectral Set Conjecture, for all sets A of size |A| ≤ 5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd.

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 , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon,...

Factoring an odd abelian group by lacunary cyclic subsets

Sándor Szabó (2010)

Discussiones Mathematicae - General Algebra and Applications

It is a known result that if a finite abelian group of odd order is a direct product of lacunary cyclic subsets, then at least one of the factors must be a subgroup. The paper gives an elementary proof that does not rely on characters.

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