A zero-sum-free sequence of multiplicity $\le M$ in a finite abelian group has length $< 3\sqrt{M|G|}$

Let $G$ be a finite abelian group, let $M \ge 1$, and let $A$ be a sequence (a multiset) of $k$ elements of $G$ in which no element of $G$ occurs more than $M$ times. Assume $A$ is zero-sum-free, that is, no nonempty subsequence of $A$ sums to $0$:

unless every $\varepsilon_a$ is $0$. Then

This is Theorem 3 of Erdős, Lev, Rauzy, Sándor and Sárközy, and it is the general result from which they deduce the bound $F(N) < 3\sqrt{N}+1$ for non-dividing subsets of $\{1,\ldots,N\}$: one takes $G = \mathbb{Z}/a\mathbb{Z}$ for $a$ the least element of the set. Taking $M = 1$ recovers the statement that a zero-sum-free set in a finite abelian group has fewer than $3\sqrt{|G|}$ elements. The constant $3$ cannot be replaced by anything below $\sqrt{2}$, as the example $A = \{1, 2, \ldots, k\} \subseteq \mathbb{Z}/n\mathbb{Z}$ with $k \le \sqrt{2n}-1$ shows; the authors ask for the optimal constant as their Problem 7.

Formalization Note The sequence is a Multiset G, the multiplicity hypothesis is stated with Multiset.count, and subsequences are sub-multisets $S \le A$, with $S \neq 0$ meaning $S$ is nonempty. The hypothesis $M \ge 1$ is needed only to exclude the degenerate reading in which both $A$ and $M$ are empty/zero, where the right-hand side would be $0$.