Zero-sum subsequences in bounded-sum $\{-1,1\}$-sequences

The following result gives the flavor of this paper: Let $t$, $k$ and $q$ be integers such that $q\ge 0$, $0\le t<k$ and $t\equiv k(\mathrm{m}\mathrm{o}\mathrm{d}2)$, and let $s\in [0,t+1]$ be the unique integer satisfying $s \equiv q + \frac{k-t-2}{2} ,({\rm mod} , (t+2))$\mathrm{.}Thenforanyinteger$n$ such that $$n\ge \max \{k,\frac{1}{2(t+2)}{k}^2+\frac{q-s}{t+2}k-\frac{t}{2}+s\}$$ and any function $f:[n]\to \{-1,1\}$ with $|\sum_{i=1}^nf(i)| \le q$,thereisaset$B \subseteq [n]$of$k$consecutiveintegerswith$|\sum_{y\in B}f(y)| \le t$. Moreover, this bound is sharp for all the parameters involved and a characterization of the extremal sequences is given. This and other similar results involving different subsequences are presented, including decompositions of sequences into subsequences of bounded weight.