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.

我们研究了由Caro，Hansberg和Montejano引入的在有界和序列中查找零和块的问题。卡罗等。确定最小值$\{-\mathrm{1个}，\mathrm{1个}\}$-当存在k个连续项之和为零时的序列长度。我们确定集合时对应的最小序列长度$\{-\mathrm{1个}，\mathrm{1个}\}$ 被替换为 $\{-\mathrm{[R}，s\}$对于任意正整数r和s。这证实了他们的猜想。我们还构造$\{-\mathrm{1个}，\mathrm{1个}\}$-以k为单位的二次方长度序列，可避免由算术级数加总为零的k个项。这解决了他们的第二个猜想$\{-\mathrm{1个}，\mathrm{1个}\}$零和算术子序列上的-序列。最后，对于最小的序列长度，我们给了足够大的k一个超线性下界，以求出一般的零和算术级数$\{-\mathrm{[R}，s\}$-序列。