Exponential Lower Bounds on the Generalized Erd\H{o}s-Ginzburg-Ziv Constant

For a finite abelian group $G$, the generalized Erd\H{o}s--Ginzburg--Ziv constant $\mathsf s_{k}(G)$ is the smallest $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. If $n = \exp(G)$ is the exponent of $G$, the previously best known bounds for $\mathsf s_{kn}(C_n^r)$ were linear in $n$ and $r$ when $k\ge 2$. Via a probabilistic argument, we produce the exponential lower bound $$\mathsf s_{2n}(C_n^r) > \frac{n}{2}[1.25 - O(n^{-3/2})]^r$$ for $n > 0$. For the general case, we show $$\mathsf s_{kn}(C_n^r) > \frac{kn}{4}\Big(1+\frac{1}{ek} + O\Big(\frac{1}{n}\Big)\Big)^r.$$