A characterization of sequences with the minimum number of $k$-sums modulo $k$
Let $G$ be an additive abelian group of order $k$, and $S$ be a sequence over $G$ of length $k+r$, where $1\le r\le k-1$. We call the sum of $k$ terms of $S$ a $k$-sum. We show that if $0$ is not a $k$-sum, then the number of $k$-sums is at least $r+2$ except for $S$ containing only two distinct elements, in which case the number of $k$-sums equals $r+1$. This result improves the Bollobás–Leader theorem, which states that there are at least $r+1$ $k$-sums if 0 is not a $k$-sum.