A generalization of sumset and its applications

Ontology type: schema:ScholarlyArticle

Article Info

DATE

2018-10-25

AUTHORS ABSTRACT

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by h(r)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{(r)}A$$\end{document}, is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for h(r)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{(r)}A$$\end{document} is to find a lower bound for |h(r)A|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|h^{(r)}A|$$\end{document} in terms of |A|. The inverse problem for h(r)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{(r)}A$$\end{document} is to determine the structure of the finite set A for which |h(r)A|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|h^{(r)}A|$$\end{document} is minimal with respect to some fixed value of |A|. If G=Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = \mathbb {Z}$$\end{document}, the direct and inverse problems are well studied. In case of G=Z/pZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = \mathbb {Z}/p\mathbb {Z}$$\end{document}, p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset h(r)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{(r)}A$$\end{document} in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case G=Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = \mathbb {Z}$$\end{document}. More... »

PAGES

55

Journal

TITLE

Proceedings - Mathematical Sciences

ISSUE

5

VOLUME

128

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http://scigraph.springernature.com/pub.10.1007/s12044-018-0437-9

DOI

http://dx.doi.org/10.1007/s12044-018-0437-9

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https://app.dimensions.ai/details/publication/pub.1107849345

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