On zero sum subsequences of restricted size

Abstract

Let G be a finite abelian group with exp(G) = e. Let s(G) be the minimal integer t with the property that any sequence of t elements in G contains an e-term subsequence with sum zero. Let n, mand r be positive integers and m ≥ 3. Furthermore, η(C ^r _m ) = a _r (m − 1) + 1, for some constant a _r depending on r and n is a fixed positive integer such that

$$ n \geqslant \frac{{m^r (c(r)m - a_r (m - 1) + m - 3)(m - 1) - (m + 1) + (m + 1)(a_r + 1)}} {{m(m + 1)(a_r + 1)}} $$

and s(C ^r _n ) = (a _r +1)(n−1)+1. In the above lower bound on n, c(r) is the Alon-Dubiner constant. Then s(C ^r _nm ) = (a _r + 1)(nm − 1) + 1.