math.NT — Number Theory

Small-index subsequences in finite cyclic groups

Contributed by Chao Liu

For $r\ge1$, let $I_r(n)$ be the least length forcing a nonempty zero-sum subsequence of index at most $r$ over the cyclic group $C_n$. We prove $I_r(n)\le n^{1+o(1)}$ when $r\ge C[\omega(n)\log(2\omega(n))]^2$, and $I_1(p^k)\ll(k+1)p^k\log(2p^k)$ for prime powers. We also construct sequences showing that $I_r(n)/n$ is unbounded for every fixed $r$; for $r=1$, all terms may be generators.