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Subsequence sums of zero-sum free sequences over finite abelian groups

Yongke Qu, Xingwu Xia, Lin Xue, Qinghai Zhong (2015)

Colloquium Mathematicae

Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that | Σ ( X ) | 2 r - 1 ( | X | - r + 2 ) - 1 for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.

Substitution invariant sturmian bisequences

Bruno Parvaix (1999)

Journal de théorie des nombres de Bordeaux

We prove that a Sturmian bisequence, with slope α and intercept ρ , is fixed by some non-trivial substitution if and only if α is a Sturm number and ρ belongs to ( α ) . We also detail a complementary system of integers connected with Beatty bisequences.

Substitutions, abstract number systems and the space filling property

Clemens Fuchs, Robert Tijdeman (2006)

Annales de l’institut Fourier

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

Substitutions par des motifs en dimension 1

N. Pytheas Fogg (2007)

RAIRO - Theoretical Informatics and Applications

Une substitution est un morphisme de monoïdes libres : chaque lettre a pour image un mot, et l'image d'un mot est la concaténation des images de ses lettres. Cet article introduit une généralisation de la notion de substitution, où l'image d'une lettre n'est plus un mot mais un motif, c'est-à-dire un “mot à trous”, l'image d'un mot étant obtenue en raccordant les motifs correspondant à chacune de ses lettres à l'aide de règles locales. On caractérise complètement les substitutions par des motifs...

Substitutions with Cofinal Fixed Points

Bo TAN, Zhi-Xiong WEN, Jun WU, Zhi-Ying WEN (2006)

Annales de l’institut Fourier

Let ϕ be a substitution over a 2-letter alphabet, say { a , b } . If ϕ ( a ) and ϕ ( b ) begin with a and b respectively, ϕ has two fixed points beginning with a and b respectively.We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

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