Freiman, Gregory A.; Stanchescu, Yonutz V. Sets with several centers of symmetry. (English) Zbl 1231.11027 Int. J. Number Theory 7, No. 5, 1115-1135 (2011). Given \( \{ b_0, b_1, \dots, b_{s-1} \} \subseteq \mathbb{Z} ^2.\) For \(A\subseteq \mathbb{Z}^2\), define \[ D_i=D_i(A)=\{a-a': a\in A, a'\in A, a+a'=b_i\}, \]\[ \text{Diff}_s(A)=D_0 \cup D_1 \cup \cdots \cup D_{s-1},\quad R_s(A)=|\text{Diff}_s(A)|. \] For \(s=1,2,3,\) the maximal values of \(R_s(A)\) have been studied in [Funct. Approximatio, Comment. Math. 37, Part 1, 131–148 (2007; Zbl 1210.11106), ibid. 41, No. 2, 167–183 (2009; Zbl 1211.11110)].In this paper, the authors prove the following theorem: Let \(A\) be a finite subset of \(\mathbb{Z}^2\) with \(|A|=k.\) If \(k\) is sufficiently large and if \(b_0=(0, 0), \, b_1=(1, 0), \, b_2=(0, 1), \, b_3=(1, 1),\) then \(R_4(A)\leq 4k-\sqrt{8k+1}.\) Moreover, the equality \( R_4(A)= 4k -\sqrt{8k+1}\) holds if and only if there is \(t\in \mathbb{Z} \) such that \(k=t(2t-1)\) and \(A=\{ (x,y) \in \mathbb{Z}^2: |x-\frac{1}{4}|+ |y-\frac{1}{4}|< t \}\). Reviewer: Yong-Gao Chen (Nanjing) MSC: 11B75 Other combinatorial number theory 11P70 Inverse problems of additive number theory, including sumsets 52C99 Discrete geometry 05D99 Extremal combinatorics Keywords:inverse additive number theory; Kakeya problem; symmetry; additive combinatorics Citations:Zbl 1210.11106; Zbl 1211.11110 PDFBibTeX XMLCite \textit{G. A. Freiman} and \textit{Y. V. Stanchescu}, Int. J. Number Theory 7, No. 5, 1115--1135 (2011; Zbl 1231.11027) Full Text: DOI References: [1] DOI: 10.7169/facm/1229618746 · Zbl 1210.11106 · doi:10.7169/facm/1229618746 [2] DOI: 10.7169/facm/1261157808 · Zbl 1211.11110 · doi:10.7169/facm/1261157808 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.