Bounds on an additive combinatorics problem (just looking for references)

Question

I'm looking for known results of a problem, but i don't know the right terms to look for.

What is the minimal number $a$, s.t. any subset $A\subseteq \mathbb{Z}_3^n$ $\ \ \ |A|\ge a$, contains 3 distinct elements $x,y,z$ satisfying $x+y+z=0$

Answer 1

I think the term you are looking for is "zero sums", or similar variants. See e.g. This article.

Theorem 2.5 seems relevant. It is attributed to Harboth, and states that if $f(m,d)$ is the minimal length of a sequence necessary to guarantee a zero-sum in $\mathbb Z_m^d$, then $(m-1)2^d + 1 \leq f(m,d) \leq (m-1)m^d +1$.