On Two-Dimensional Sums in Abelian Groups

A. A. Uvakin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: It is proved that if, for a subset $A$ of a finite Abelian group $G$, under the action of a linear operator $L\colon G^3 \to G^2$, the image $L(A,A,A)$ has cardinality less than $(7/4)|A|^2$, then there exists a subgroup $H \subseteq G$ and an element $x \in G$ for which $A \subseteq H+x$; further, $|H| < (3/2)|A|$.

Keywords: Abelian group, linear operator, convolution, sums of sets, additive combinatorics.

UDC: 512.541

Received: 21.07.2016
Revised: 01.03.2017

DOI: 10.4213/mzm11319

 English version: Mathematical Notes, 2018, 103:2, 271–289

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