Asymptotics for the logarithm of the number of \((k,l)\)-sum-free sets in groups of prime order (Q1759154)

A set \(A\) in a commutative group is \((k,l)\)-sum-free for integers \(k \neq l\), if the equation \(x_1+ \dots +x_k = y_1 + \dots +y_l\) has no solution with \(x_i, y_j \in A\). For \(k=2, l=1\) these are the familiar sum-free sets. The author proves that the number of \((k,l)\)-sum-free subsets of \(Z_p\) is \(2^{p/(k+l) +o(p)}\). The maximal such sets are arithmetic progressions, and the lower bound counts their subsets, which coincide only minimally. The upper bound is based on the method used by \textit{B. Green} and the reviewer to count sum-free subsets of commutative groups [Isr. J. Math. 147, 157--188 (2005; Zbl 1158.11311)].