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 minimum $\{-1,1\}$-sequence length for when there exist $k$ consecutive terms that sum to zero. corresponding sequence set $\{-1,1\}$ is replaced $\{-r,s\}$ arbitrary positive integers $r$ $s.$ This confirms a conjecture theirs. also construct $\{-1,1\}$-sequences quadratic avoid indexed an arithmetic progression solves second theirs case on subsequences. Finally, we give superlinear lower bound find general $\{-r,s\}$-sequences.