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Counting \((k,l)\)-sumsets in groups of a prime order. (English) Zbl 1354.11021
From the text: A subset \(A\) of a group \(G\) is called \((k,l)\)-sumset, if \(A=kB-lB\) for some \(B\subseteq G\), where \(kB-lB=x_1+\ldots+x_k-x_{k+1}-\ldots-x_{k+l}: x_1,\ldots,x_{k+l}\in B\). Upper and lower bounds for the number \((k,l)\)-sumsets in groups of prime order are provided.
Write \(SS_{k,l}(\mathbb Z_p)\) for the collection of \((k, l)\)-sumsets in \(\mathbb Z_p\). B. Green and I. Ruzsa in [Stud. Sci. Math. Hung. 41, No. 3, 285–293 (2004; Zbl 1064.11020)] proved
\[ p^22^{p/3} \ll | SS_{2,0}(\mathbb Z_p)| \leq 2^{p/3+\theta(p)} \]
where \(\theta(p)/p\to 0\) as \(p\to\infty\) and \(\theta(p)\ll p(\log \log p)^{2/3}(\log p)^{-1/9}\).
The aim of this work is to obtain bounds for the number \(| SS_{k,l}(\mathbb Z_p)|\). We prove
Theorem 1. Let \(p\) be a prime number and \(k,l\) be nonnegative integers with \(k + l \geq 2\). Then there exists a positive constant \(C_{k,l}\) such that
\[ C_{k,l}2^{p/(2(k+l)-1)} \leq | SS_{k,l}(\mathbb Z_p)| \leq 2^{(p/(k+l+1))+(k+l-2)+o(p)}. \tag{1} \]

11B75 Other combinatorial number theory
68R05 Combinatorics in computer science
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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