Asymptotics of the logarithm of the number of \((k, l)\)-sum-free sets in an abelian group (Q314126)

A set \(A\) in a commutative group \(G\) is called \((k,l)\)-sumfree, if the equation \(x_1 + \ldots +x_k = y_1 + \ldots +y_l\) has no solution in \(A\). For a finite group, \(\mu_{k,l}(G)\) denotes the maximal size of a \((k,l)\)-sumfree set, and \(SF_{k,l}(G)\) the number of all such sets. Applying a method of \textit{B. Green} and the reviewer [Isr. J. Math. 147, 157--188 (2005; Zbl 1158.11311)], the author proves the logarithmic asymptotic formula NEWLINE\[NEWLINE \log_2 | SF_{k,l}(G) | = \mu_{k,l}(G) + o( |G| ). NEWLINE\]NEWLINE Heuristically this means that most such sets are subsets of a few maximal \((k,l)\)-sumfree sets.NEWLINENEWLINEGood estimates of \(\mu_{k,l}(G)\) are due to \textit{B. Bajnok} [Int. J. Number Theory 5, No. 6, 953--971 (2009; Zbl 1231.11123)] and the author [Diskretn. Anal. Issled. Oper. 20, No. 3, 45--64 (2013); translation in J. Appl. Ind. Math. 7, No. 4, 574--587 (2013; Zbl 1324.11026)], but the exact value is not known for all groups.