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BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

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


In this talk, we consider problems and results that go in the opposite direction of the classical theorems in the discrepancy theory. The following statement gives a flavor of our approach. Let $t$, $k$ and $q$ be integers such that $q \ge 0$, $0 \le t < k$, and $t \equiv k \,({\rm mod}\, 2)$, and take $s \in [0,t+1]$ as the unique integer satisfying $s \equiv q + \frac{k-t-2}{2} \,({\rm mod} \, (t+2))$. Then, for any integer $$n \ge \frac{1}{2(t+2)}k^2 + \frac{q-s}{t+2}k - \frac{t}{2} + s$$ and any function $f:[n]\to \{-1,1\}$ with $|\sum_{i=1}^nf(i)|\leq q$, there is a block of $k$ consecutive terms ($k$-block) $B\subset [n]$ with $|\sum_{x\in B}^nf(x)|\leq t$. Moreover, this bound is sharp for all the parameters involved and a characterization of the extremal sequences is given. This is a joint work with Yair Caro and Adriana Hansberg.

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