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 for sufficiently large k a superlinear lower bound on the minimum sequence length to find a zero-sum arithmetic progression for general {−r,s}-sequences.