Zero-sum subsequences in bounded-sum {−1,1}-sequences

Abstract

The following result gives the flavor of this paper: Let t, k and q be integers such that q≥0, 0≤t<k and t≡k(mod2), and let s∈[0,t+1] be the unique integer satisfying s≡q+k−t−22(mod(t+2)). Then for any integer n such thatn≥max⁡{k,12(t+2)k2+q−st+2k−t2+s} and any function f:[n]→{−1,1} with |∑i=1nf(i)|≤q, there is a set B⊆[n] of k consecutive integers with |∑y∈Bf(y)|≤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.