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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
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