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 \[ \mathsf s_{kn}(C_n^r) > \frac{kn}{4}\Big(1+\frac{1}{ek} + O\Big(\frac{1}{n}\Big)\Big)^r. \]

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广义 Erdős-Ginzburg-Ziv 常数的指数下界

对于有限阿贝尔群 $G$，广义 Erd\H{o}s--Ginzburg--Ziv 常数 $\mathsf s_{k}(G)$ 是最小的 $m$，使得 $m$ 的序列$G$ 中的元素总是包含一个 $k$-element 子序列，其总和为零。如果 $n = \exp(G)$ 是 $G$ 的指数，则 $\mathsf s_{kn}(C_n^r)$ 之前最知名的边界在 $n$ 和 $r$ 中是线性的，当 $k \ge 2$。通过概率论，我们产生指数下界 \[ \mathsf s_{2n}(C_n^r) > \frac{n}{2}[1.25 - O(n^{-3/2})]^r \] 表示 $n > 0$。对于一般情况，我们显示 \[ \mathsf s_{kn}(C_n^r) > \frac{kn}{4}\Big(1+\frac{1}{ek} + O\Big(\frac{ 1}{n}\Big)\Big)^r。\]