Result: The number of k-sums of abelian groups of order k: The number of \(k\)-sums of Abelian groups of order \(k\)

Let \(G\) be an Abelian group of order \(k\) and let \(r\geq 1\). Consider sequences \(a_1, \dots , a_{k+r}\) of elements of \(G\) with the property that no sum of \(k\) distinct terms is 0. It is proved, confirming a conjecture of \textit{B. Bollobás} and \textit{I. Leader} [J. Number Theory 78, No. 1, 27--35 (1999; Zbl 0929.11008)] that among such sequences the one for which the cardinality of different values of \(k\)-fold sums is minimal has the form \(b_1, \dots , b_{r+1}, 0, \dots , 0\), where the \(b_i\)'s are chosen to minimize the number of sums without 0 being a nonempty sum. Bollobás and Leader (ibid.) proved that this cardinality is at least \(r+1\) for \(r\leq k-1\); the author describes the cases of equality in this bound.