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 $q\geq 0$, $0\leq t < k$ and $t \equiv k \,({\rm mod}\, 2)$, and let $s\in [0,t+1]$ be the unique integer satisfying $s \equiv q + \frac{k-t-2}{2} \,({\rm mod} \, (t+2))$. Then for any integer $n$ such that \[n \ge \max\left\{k,\frac{1}{2(t+2)}k^2 + \frac{q-s}{t+2}k - \frac{t}{2} + s\right\}\] and any function $f:[n]\to \{-1,1\}$ with $|\sum_{i=1}^nf(i)| \le q$, there is a set $B \subseteq [n]$ of $k$ consecutive integers with $|\sum_{y\in B}f(y)| \le t$. 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. 
Yair Caro, Adriana Hansberg and Amanda Montejano
math.CO math.NT