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 Diskretn. Anal. Issled. Oper., 2018, Volume 25, Number 4, Pages 97–111 (Mi da911)

The number of $k$-sumsets in an Abelian group

A. A. Sapozhenko, V. G. Sargsyan

Lomonosov Moscow State University, 1 Leninskie gory, 119991 Moscow, Russia

Abstract: Let $G$ be an Abelian group of order $n$. The sum of subsets $A_1,…,A_k$ of $G$ is defined as the collection of all sums of $k$ elements from $A_1,…,A_k$; i.e., $A_1+…+A_k=\{a_1+…+a_k\mid a_1\in A_1,…, a_k\in A_k\}$. A subset representable as the sum of $k$ subsets of $G$ is a $k$-sumset. We consider the problem of the number of $k$-sumsets in an Abelian group $G$. It is obvious that each subset $A$ in $G$ is a $k$-sumset since $A$ is representable as $A=A_1+…+ A_k$, where $A_1=A$ and $A_2=…=A_k=\{0\}$. Thus, the number of $k$-sumsets is equal to the number of all subsets of $G$. But, if we introduce a constraint on the size of the summands $A_1,…,A_k$ then the number of $k$-sumsets becomes substantially smaller. A lower and upper asymptotic bounds of the number of $k$-sumsets in Abelian groups are obtained provided that there exists a summand $A_i$ such that $|A_i|\geq n\log^qn$ and $|A_1+…+A_{i-1}+ A_{i+1}+…+A_k|\geq n\log^qn$, where $q=- 1/8$ and $i\in\{1,…,k\}$. Bibliogr. 8.

Keywords: set, characteristic function, group, progression, coset.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00593а The authors were supported by the Russian Foundation for Basic Research (project no. 16-01-00593a).

DOI: https://doi.org/10.17377/daio.2018.25.608

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English version:
Journal of Applied and Industrial Mathematics, 2018, 12:4, 729–737

Document Type: Article
UDC: 519.1
Revised: 13.06.2018

Citation: A. A. Sapozhenko, V. G. Sargsyan, “The number of $k$-sumsets in an Abelian group”, Diskretn. Anal. Issled. Oper., 25:4 (2018), 97–111; J. Appl. Industr. Math., 12:4 (2018), 729–737

Citation in format AMSBIB
\Bibitem{SapSar18} \by A.~A.~Sapozhenko, V.~G.~Sargsyan \paper The number of $k$-sumsets in an Abelian group \jour Diskretn. Anal. Issled. Oper. \yr 2018 \vol 25 \issue 4 \pages 97--111 \mathnet{http://mi.mathnet.ru/da911} \crossref{https://doi.org/10.17377/daio.2018.25.608} \elib{http://elibrary.ru/item.asp?id=36449713} \transl \jour J. Appl. Industr. Math. \yr 2018 \vol 12 \issue 4 \pages 729--737 \crossref{https://doi.org/10.1134/S1990478918040130} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85058094103}