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
$A_1,\dots,A_k$ of
$G$ is defined as the collection of all sums of
$k$ elements from
$A_1,\dots,A_k$; i.e., $A_1+\dots+A_k=\{a_1+\dots+a_k\mid a_1\in A_1,\dots, a_k\in A_k\}$. 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
$A=A_1+\dots+ A_k$, where
$A_1=A$ and
$A_2=\dots=A_k=\{0\}$. 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
$A_1,\dots,A_k$ 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 $|A_1+\dots+A_{i-1}+ A_{i+1}+\dots+A_k|\geq n\log^qn$, where
$q=- 1/8$ and
$i\in\{1,\dots,k\}$. Bibliogr. 8.
Keywords:
set, characteristic function, group, progression, coset.
UDC:
519.1 Received: 29.01.2018
Revised: 13.06.2018
DOI:
10.17377/daio.2018.25.608
Fulltext:
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