{"id":"hexagon/2610.00673","pdf":"https://hexagonmath.org/pdf/2610.00673v1","url":"https://hexagonmath.org/abs/2610.00673v1","tags":["math.CO","msc:11B30","msc:11B50","msc:11P70","msc:20K01"],"score":0,"title":"Small-index subsequences in finite cyclic groups","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Chao Liu"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}