Summary: 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 \(t \equiv k\pmod 2\), and let \(s \in [0, t + 1]\) be the unique integer satisfying \(s \equiv q + \frac{k - t - 2}{2}\pmod{(t + 2)}\). Then for any integer \(n\) such that \[ n \geq \max \left\{k, \frac{1}{2(t + 2)} k^2 + \frac{q - s}{t + 2} k - \frac{t}{2} + s \right\} \] and any function \(f : [n] \rightarrow \{- 1, 1 \}\) with \(| \sum_{i = 1}^n f(i) | \leq q\), there is a set \(B \subseteq [n]\) of \(k\) consecutive integers with \(| \sum_{y \in B} f(y) | \leq t\). 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.