Estimates related to sumfree subsets of sets of integers (Q1355251)

A set \(A\) of integers is sumfree if \(A\cap (A+A)=\emptyset \). For any set \(B\) of positive integers and any real \(x\) the set \( A_x = \{ n\in B: 1/3 < \{nx\} < 2/3 \} \) is an example of a sumfree subset of \(B\). Hence, as \textit{P. Erdös} [Proc. Symp. Pure Math. 8, 181-189 (1965; Zbl 0144.28103)] and \textit{N. Alon} and \textit{D. J. Kleitman} in [A tribute to Paul Erdös, Cambridge Univ. Press, 13-26 (1990; Zbl 0718.11006)] observed, an easy averaging yields \( |A_x |\geq {1\over 3} ( |B |+1) \) for some \(x\). This bound is improved by 1 for \( |B |\equiv 1 \pmod{3} \) by analyzing the Fourier expansion of the function \( |A_x |\). The analogous problem with the condition \(A\cap (A+ ... +A)=\emptyset \), \(k\) summands, is also considered. Denoting the size of the maximal subset of \(B\) with this property by \(s_k(B)\), the analogous obvious bound \( |B |/4\) for \(s_4(B)\) is improved by \( c ( \log |B |) / \log\log |B |\). Finally examples are constructed that satisfy \(s_k(B)<\delta _k |B |\) with certain \(\delta _k\rightarrow 0\).