Volume 25, No 4, 2018, P. 97-111

UDC 519.1
A. A. Sapozhenko and V. G. Sargsyan
The number of $k$-sumsets in an Abelian group

Abstract:
Let $G$ be an Abelian group of order $n$. The sum of subsets $A_1, \ldots , A_k$ of $G$ is defined as the collection of all sums of $k$ elements from $A_1, \ldots , A_k$; i. e., $A_1 + \cdots + A_k = {a_1 + \cdots + a_k | a_1 \in A_1, \ldots , a_k \in A_k}$. 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 $A = A_1 + \cdots + A_k$, where $A_1 = A$ and $A_2 = \cdots = A_k = {0}$. 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 $A_1, \ldots , A_k$ 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 $|A_i| \ge n$ log$^{q} n$ and $|A_1 + \cdots + A_{i - 1} + A_{i+1} + \cdots + A_k| \ge n$ log$^{q} n$, where $q = -1/8$ and $i \in {1, \ldots , k}$.
Bibliogr. 8.

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

DOI: 10.17377/daio.2018.25.608

Aleksandr A. Sapozhenko ¹
Vahe G. Sargsyan ¹
1. Lomonosov Moscow State University,
1 Leninskie Gory, 119991 Moscow, Russia
e-mail: sapozhenko@mail.ru, vahe_sargsyan@ymail.com

Received 29 January 2018
Revised 13 June 2018

References