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

Abstract: 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.