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Title: | A generalization of sumset and its applications |

Authors: | Mistri R.K. Pandey R.K. Prakash O. |

Published in: | Proceedings of the Indian Academy of Sciences: Mathematical Sciences |

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, 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 is to find a lower bound for | h( r )A| in terms of |A|. The inverse problem for h( r )A is to determine the structure of the finite set A for which | h( r )A| is minimal with respect to some fixed value of |A|. If G= Z, the direct and inverse problems are well studied. In case of G= Z/ pZ, p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271â€“279). In this paper, we express the generalized sumset h( r )A 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. Â© 2018, Indian Academy of Sciences. |

Citation: | Proceedings of the Indian Academy of Sciences: Mathematical Sciences (2018), 128(5): - |

URI: | https://doi.org/10.1007/s12044-018-0437-9 http://repository.iitr.ac.in/handle/123456789/9831 |

Issue Date: | 2018 |

Publisher: | Springer |

Keywords: | Arithmetic progression direct and inverse problems h-fold sumsets |

ISSN: | 2534142 |

Author Scopus IDs: | 56203915400 35097679700 57210722800 |

Author Affiliations: | Mistri, R.K., Department of Mathematics, Harish-Chandra Research Institute, Allahabad, 211 019, India Pandey, R.K., Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247 667, India Prakash, O., Department of Mathematics, Indian Institute of Technology Patna, Patna, 800 013, India |

Corresponding Author: | Pandey, R.K.; Department of Mathematics, Indian Institute of Technology RoorkeeIndia; email: ramkpandey@gmail.com |

Appears in Collections: | Journal Publications [MA] |

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