Geodesic
The number of $k$-sumsets in an Abelian group
Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 4, pp. 97-111.

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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, coset.
Mots-clés : group, progression 
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A. A. Sapozhenko; V. G. Sargsyan. The number of $k$-sumsets in an Abelian group. Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 4, pp. 97-111. http://geodesic.mathdoc.fr/item/DA_2018_25_4_a6/

[1] V. G. Sargsyan, “The number of differences in groups of prime order”, Discrete Math. Appl., 23:2 (2013), 195–201 | DOI | DOI | MR | Zbl

[2] V. G. Sargsyan, “Counting sumsets and differences in abelian group”, Diskretn. Anal. Issled. Oper., 22:2 (2015), 73–85 (Russian) | DOI | MR | Zbl

[3] Alon N., “Large sets in finite fields are sumsets”, J. Number Theory, 126 (2007), 110–118 | DOI | MR | Zbl

[4] Alon N., Granville A., Ubis A., “The number of sumsets in a finite field”, Bull. Lond. Math. Soc., 42:5 (2010), 784–794 | DOI | MR | Zbl

[5] Croot E., Lev V. F., “Open problems in additive combinatorics”, Additive Combinatorics, CRM Proc. Lect. Notes, 43, Amer. Math. Soc., Providence, RI, 2007, 207–233 | DOI | MR | Zbl

[6] Green B., Essay submitted for the Smith's Prize, Camb. Univ., Cambridge, 2001

[7] Green B., Ruzsa I. Z., “Counting sumsets and sum-free sets modulo a prime”, Stud. Sci. Math. Hung., 41:3 (2004), 285–293 | MR | Zbl

[8] Green B., Ruzsa I. Z., “Sum-free sets in abelian groups”, Isr. J. Math., 147 (2005), 157–188 | DOI | MR | Zbl