Zero-sum subsequences in bounded-sum \{-r,s\}-sequences

We study the problem of finding zero-sum blocks in bounded-sum sequences, which was introduced by Caro, Hansberg, and Montejano. Caro et al. determine the minimum $\{-1,1\}$-sequence length for when there exist $k$ consecutive terms that sum to zero. We determine the corresponding minimum sequence length when the set $\{-1,1\}$ is replaced by $\{-r,s\}$ for arbitrary positive integers $r$ and $s.$ This confirms a conjecture of theirs. We also construct $\{-1,1\}$-sequences of length quadratic in $k$ that avoid $k$ terms indexed by an arithmetic progression that sum to zero. This solves a second conjecture of theirs in the case of $\{-1,1\}$-sequences on zero-sum arithmetic subsequences. Finally, we give a superlinear lower bound on the minimum sequence length to find a zero-sum arithmetic progression for general $\{-r,s\}$-sequences.