Displaying similar documents to “Some results in additive number theory I: The critical pair theory”

A characterization of sequences with the minimum number of k-sums modulo k

Xingwu Xia, Yongke Qu, Guoyou Qian (2014)

Colloquium Mathematicae

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Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

Restricted set addition in Abelian groups: results and conjectures

Vsevolod F. Lev (2005)

Journal de Théorie des Nombres de Bordeaux

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We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed. The paper assumes no expertise from the reader and can serve as an introduction to the subject.

On abelian versions of critical factorization theorem

Sergey Avgustinovich, Juhani Karhumäki, Svetlana Puzynina (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.