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Counting sumsets and sum-free sets modulo a prime. (English) Zbl 1064.11020
Let $$p$$ be a prime number and let $$\mathbb Z_p$$ be the group of residues modulo $$p$$. $$A$$ subset $$A$$ of $$\mathbb Z_p$$ is called a sumset, if there are $$A=B+ B=\{b+b'\mid b,b'\in B\subseteq \mathbb Z_p\}$$. Let $$| (SS(\mathbb Z_p)|$$ denote the cardinality of the family of all sumsets of $$Z_p$$ . A subset $$A$$ of $$Z_p$$ is sum-free, if there are no solutions to $$a=a'+a''$$ with $$a,a',a''\in A$$. Write $$| SF(\mathbb Z_p)|$$ for the cardinality of the collection of sum-free subsets of $$\mathbb Z_p$$. The authors show that $| SF(\mathbb Z_p)| \leq 2^{\frac p3+ \kappa(p)}\quad\text{and}\quad p^2 2^{\frac p3}\ll| SS(\mathbb Z_p)| \leq 2^{\frac p3+\kappa(p)}$ where $$\kappa(p)\to 0$$ as $$p\to \infty$$. The first inequality $$| SF(\mathbb Z_p)| \leq 2^{\frac p3+\kappa(p)}$$ improves the upper bound of V. F. Lev and T. Schoen [Finite Fields Appl. 8., No. 1, 108–119 (2002; Zbl 0997.11020)].

##### MSC:
 11B13 Additive bases, including sumsets 11P70 Inverse problems of additive number theory, including sumsets
##### Keywords:
sumset; sum-free set; granularization
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