NUMBER OF WEIGHTED SUBSEQUENCE SUMS WITH WEIGHTS IN {1, ?1}

摘要

Let G be an abelian group of order n and let it be of the form G ～= Zn1 ⊕ Zn2 ⊕ · · · ⊕ Znr , where ni | ni+1 for 1 ≤ i < r and n1 > 1. Let A = {1, ?1}. Given a sequence S with elements in G and of length n + k such that the natural number k satisfies k ≥ 2r! ?1 ? 1 + r! 2 , where r" = |{i ∈ {1, 2, · · · , r} : 2 | ni}|, if S does not have an A-weighted zero-sum subsequence of length n, we obtain a lower bound on the number of A-weighted n-sums of the sequence S. This is a weighted version of a result of Bollob′as and Leader. As a corollary, one obtains a result of Adhikari, Chen, Friedlander, Konyagin and Pappalardi. A result of Yuan and Zeng on the existence of zero-smooth subsequences and the DeVos-Goddyn-Mohar Theorem are some of the main ingredients of our proof.