A generalization of an addition theorem of Shatrowsky
type: Item
instance of: scholarly article
title: A generalization of an addition theorem of Shatrowsky
zbMATH Open document ID: 0768.05051
DOI: 10.1016/S0195-6698(05)80031-9
author: Yahya Ould Hamidoune
published in: European Journal of Combinatorics
publication date: 1993-01-16T00:00:00Z
review text: The Cayley-Davenport theorem states that if \(A\) and \(B\) are proper subsets of \(\mathbb Z_n\) with \(A+B\neq\mathbb Z_n\), then \(| A+B|\geq| A|+| B|-1\) for a prime \(n\). Chowla generalized this result to composite \(n\) and a further generalization was obtained by Shatrowsky [\textit{L. I. Shatrovskiĭ}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 45, 315--317 (1944; Zbl 0063.06948)]. After a thorough introduction to directed graphs, the author uses Cayley graphs to generalize Shatrowsky's theorem as follows: Let \(A\) and \(B\) be two proper finite subsets of a group \(G\) such that \(A\neq AB\cup A\langle B\rangle\). Then \(| A\cup AB|\geq| A|+\min\{| B|,\;\nu(B)\}\) where \(\langle B\rangle\) denotes the subgroup generated by \(B\) and \(\nu(B)\) is the minimal order of an element of \(B\).
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 91734
zbMATH Keywords: addition theorem
zbMATH Keywords: Cayley diagram
zbMATH Keywords: Cayley graphs
zbMATH Keywords: Shatrowsky's theorem
reviewed by: Steven C. Althoen
MaRDI profile type: Publication
cites work: On the Addition of Residue Classes
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full work available at URL: s0195-6698(05)80031-9
OpenAlex ID: W2028226840
description: scientific article; zbMATH DE number 91734