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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
$C_{k,l}2^{p/(2(k+l)-1)} \leq | SS_{k,l}(\mathbb Z_p)| \leq 2^{(p/(k+l+1))+(k+l-2)+o(p)}. \tag{1}$

##### MSC:
 11B75 Other combinatorial number theory 68R05 Combinatorics in computer science 20D60 Arithmetic and combinatorial problems involving abstract finite groups
##### Keywords:
characteristic function; sumsets; granular set