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Sets with several centers of symmetry. (English) Zbl 1231.11027

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 \(A=\{ (x,y) \in \mathbb{Z}^2: |x-\frac{1}{4}|+ |y-\frac{1}{4}|< t \}\).

MSC:

11B75 Other combinatorial number theory
11P70 Inverse problems of additive number theory, including sumsets
52C99 Discrete geometry
05D99 Extremal combinatorics
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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
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